Abstract Algebra Practice Set for Higher Mathematics Exams

This post brings together selected questions from NBHM and GATE exams focused on Abstract Algebra, providing a solid practice base for students preparing for higher-level competitive examinations in Advanced Mathematics. Each question is paired with an answer key, helping learners strengthen concepts like groups, rings, fields and polynomial structures. These topics are essential for exams involving Mathematics Coaching, Online Math Learning, and Higher Education. It also benefits aspirants preparing for Government Exams and Scholarship Tests. The curated questions ensure clarity, concept revision and deeper understanding for achieving strong scores in upcoming advanced-level mathematics examinations.

1).

Find the value of \(a \in \mathbb{Z}\) such that \(2 + \sqrt{3}\) is a root of the polynomial

\[ x^3 - 5x^2 + ax - 1 \]

Answer: \(5\)

2).

What is the number of groups of order \(6\) (upto isomorphism)?

Answer: \(2\)

3).

Let \(G\) be a cyclic group of order \(10\). For \(a \in G\), let \(\langle a \rangle\) denote the subgroup generated by \(a\). How many elements are there in the set

\[ \{a \in G \mid \langle a \rangle = G \}? \]

Answer: \(4\)

4).

Let \(\alpha = 2^{\frac{1}{3}}\) and \(\beta = 5^{\frac{1}{4}}\). Let \(L\) be the field obtained by adjoining \(\alpha\) and \(\beta\) to \(\mathbb{Q}\). What is the degree of the extension \([L : \mathbb{Q}]\)?

Answer: \(12\)

5).

Pick out the units in \(\mathbb{Z}[\sqrt{3}]\).

1. \(-7 + 4\sqrt{3}\)
2. \(5 + 3\sqrt{3}\)
3. \(2 - \sqrt{3}\)
4. \(-3 - 2\sqrt{3}\)

Answer: (a), (c)

6).

Pick out the integral domains from the following list of rings:

1. \(\{a + b\sqrt{5} \mid a,b \in \mathbb{Q}\}\)
2. The ring of continuous functions from \([0,1]\) into \(\mathbb{R}\)
3. The ring of complex analytic functions on the disc \(\{z \in \mathbb{C} \mid |z|<1 li="">
4. The polynomial ring \(\mathbb{Z}[x]\)

Answer: (a), (c), (d)

7).

Pick out the abelian groups from the following list:

1. Any group of order 4.
2. Any group of order 36.
3. Any group of order 47.
4. Any group of order 49.

Answer: (a), (c), (d)

8).

Let \( f : (\mathbb{Q}, +) \to (\mathbb{Q}, +) \) be a non-zero homomorphism. Pick out the true statements:

1. \(f\) is always one–one.
2. \(f\) is always onto.
3. \(f\) is always a bijection.
4. \(f\) need be neither one–one nor onto.

Answer: (a), (b), (c)

9).

Consider the element

\[ \alpha = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 1 & 4 & 5 & 3 \end{pmatrix} \] of the symmetric group \(S_5\) on five elements. Pick out the true statements:

1. The order of \(\alpha\) is 5.
2. \(\alpha\) is conjugate to \[ \begin{pmatrix} 4 & 5 & 2 & 3 & 1 \\ 5 & 4 & 3 & 1 & 2 \end{pmatrix}. \]
3. \(\alpha\) is the product of two cycles.
4. \(\alpha\) commutes with all elements of \(S_5\).

Answer: (b), (c)

10).

Let \(G\) be a group of order \(60\). Pick out the true statements:

1. \(G\) is abelian.
2. \(G\) has a subgroup of order \(30\).
3. \(G\) has subgroups of order \(2, 3,\) and \(5\).
4. \(G\) has subgroups of order \(6, 10,\) and \(15\).

Answer: (c)

11).

Consider the polynomial ring \(R[x]\) where \(R = \mathbb{Z}/12\mathbb{Z}\) and write the elements of \(R\) as \(\{0,1,\dots,11\}\). Write down all the distinct roots of the polynomial \(f(x)= x^2 + 7x\) of \(R[x]\).

Answer: \(0, 5, 8, 9\)

12).

Let \(R\) be the polynomial ring \(\mathbb{Z}_2[x]\) and write the elements of \(\mathbb{Z}_2\) as \(\{0,1\}\). Let \((f(x))\) denote the ideal generated by the element \(f(x) \in R\). If \(f(x) = x^2 + x + 1\), then the quotient ring \(R/(f(x))\) is

1. a ring but not an integral domain.
2. an integral domain but not a field.
3. a finite field of order 4.
4. an infinite field.

Answer: (c)

13).

Pick out the correct statements from the following list:

1. A homomorphic image of a UFD (unique factorization domain) is again a UFD.
2. The element \(2 \in \mathbb{Z}[\sqrt{-5}]\) is irreducible in \(\mathbb{Z}[\sqrt{-5}]\).
3. Units of the ring \(\mathbb{Z}[\sqrt{-5}]\) are the units of \(\mathbb{Z}\).
4. The element \(2\) is a prime element in \(\mathbb{Z}[\sqrt{-5}]\).

Answer: (b), (c)

14).

Let \(p\) and \(q\) be two distinct primes. Pick the correct statements from the following:

1. \(\mathbb{Q}(\sqrt{p})\) is isomorphic to \(\mathbb{Q}(\sqrt{q})\) as fields.
2. \(\mathbb{Q}(\sqrt{p})\) is isomorphic to \(\mathbb{Q}(\sqrt{-q})\) as vector spaces over \(\mathbb{Q}\).
3. \([\mathbb{Q}(\sqrt{p}, \sqrt{q}) : \mathbb{Q}] = 4.\)
4. \(\mathbb{Q}(\sqrt{p}, \sqrt{q}) = \mathbb{Q}(\sqrt{p} + \sqrt{q}).\)

Answer: (b), (c), (d)

15).

Let \(G\) be a group of order \(n\). Which of the following conditions imply that \(G\) is abelian?

1. \(n = 15\).
2. \(n = 21\).
3. \(n = 36\).

Answer: (a)

16).

Which of the following subgroups are necessarily normal subgroups?

1. The kernel of a group homomorphism.
2. The center of a group.
3. The subgroup consisting of all matrices with positive determinant in the group of all invertible \(n \times n\) matrices with real entries (under matrix multiplication).

Answer: (a), (b), (c)

17).

List all the units in the ring of Gaussian integers.

Answer: \(+1, -1, -i, +i\)

18).

List all possible values occurring as \(\deg f\) (degree of \(f\)) where \(f\) is an irreducible polynomial in \(\mathbb{R}[x]\).

Answer: \(1, 2\)

19).

Write down an irreducible polynomial of degree \(3\) over the field \(\mathbb{F}_3\) of three elements.

\[ x^3 + 2x + 1 \]

20).

Let \(S_7\) denote the group of permutations of 7 symbols. Find the order of the permutation:

\[ \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 6 & 4 & 5 & 7 & 3 & 1 & 2 \end{pmatrix} \]

Answer: \(6\)

21).

Write down the number of mutually nonisomorphic abelian groups of order \(19^5\).

Answer: \(7\)

22).

For two ideals \(I\) and \(J\) in a commutative ring \(R\), define \(I : J = \{ a \in R : aJ \subset I \}\). In the ring \(\mathbb{Z}\) of all integers, if \(I = 12\mathbb{Z}\) and \(J = 8\mathbb{Z}\), find \(I : J\).

Answer: \(3\mathbb{Z}\)

23).

Let \(P\) be a prime ideal in a commutative ring \(R\) and let \(S = R \setminus P\), i.e., the complement of \(P\) in \(R\). Pick out the true statements:

1. \(S\) is closed under addition.
2. \(S\) is closed under multiplication.
3. \(S\) is closed under addition and multiplication.

Answer: (b)

24).

Let \(p\) be a prime and consider the field \(\mathbb{Z}_p\). List the primes for which the following system of linear equations DOES NOT have a solution in \(\mathbb{Z}_p\):

\[ 5x + 3y = 4 \\ 3x + 6y = 1 \]

Answer: \(3\)

25).

Let \(A\) be a \(227 \times 227\) matrix with entries in \(\mathbb{Z}_{227}\), such that all its eigenvalues are distinct. Write down its trace.

Answer: \(0\)

26).

Pick out the cases where the given subgroup \(H\) is a normal subgroup of the group \(G\).

1. \(G\) is the group of all \(2 \times 2\) invertible upper triangular matrices with real entries, under matrix multiplication, and \(H\) is the subgroup of all such matrices \((a_{ij})\) such that \(a_{11} = 1\).
2. \(G\) is the group of all \(2 \times 2\) invertible upper triangular matrices with real entries, under matrix multiplication, and \(H\) is the subgroup of all such matrices \((a_{ij})\) such that \(a_{11} = a_{22}\).
3. \(G\) is the group of all \(n \times n\) invertible matrices with real entries, under matrix multiplication, and \(H\) is the subgroup of such matrices with positive determinant.

Answer: (a), (b), (c)

27).

Pick out the cases where the given subgroup \(H\) is a normal subgroup of the group \(G\).

1. \(G\) is the group of all \(2 \times 2\) invertible upper triangular matrices with real entries, under matrix multiplication, and \(H\) is the subgroup of all such matrices \((a_{ij})\) such that \(a_{11} = 1\).
2. \(G\) is the group of all \(2 \times 2\) invertible upper triangular matrices with real entries, under matrix multiplication, and \(H\) is the subgroup of all such matrices \((a_{ij})\) such that \(a_{11} = a_{22}\).
3. \(G\) is the group of all \(n \times n\) invertible matrices with real entries, under matrix multiplication, and \(H\) is the subgroup of such matrices with positive determinant.

Answer: (a), (b), (c)

28).

Let \(GL(n, \mathbb{R})\) denote the group of all invertible \(n \times n\) matrices with real entries, under matrix multiplication, and let \(SL(n, \mathbb{R})\) denote the subgroup of such matrices whose determinant is equal to unity. Identify the quotient group \(GL(n, \mathbb{R}) / SL(n, \mathbb{R})\).

Answer: multiplicative group of non-zero reals

29).

Let \(S_n\) denote the symmetric group of permutations of \(n\) symbols. Does \(S_7\) contain an element of order \(10\)? If ‘yes’, write down an example of such an element.

Answer: Yes, \((1\,2\,3\,4\,5)(6\,7)\)

30).

What is the largest possible order of an element in \(S_7\)?

Answer: \(12 \; \text{(lcm(4,3))}\)

31).

Write down all the units in the ring \(\mathbb{Z}_8\) of all integers modulo 8.

Answer: \(1, 3, 5, 7\) (numbers coprime to 8)

32).

Pick out the cases where the given ideal is a maximal ideal.

1. The ideal \(15\mathbb{Z}\) in \(\mathbb{Z}\).
2. The ideal \(I = \{f : f(0)=0\}\) in the ring \(C[0,1]\) of all continuous real-valued functions on the interval \([0,1]\).
3. The ideal generated by \(x^3 + x + 1\) in the ring of polynomials \(\mathbb{F}_3[x]\), where \(\mathbb{F}_3\) is the field of three elements.

Answer: (b)

33).

Solve the equation

\[ x^4 - 2x^3 + 4x^2 + 6x - 21 = 0 \]

given that two of its roots are equal in magnitude but opposite in sign.

Answers:
\(1 + i\sqrt{6},\; 1 - i\sqrt{6},\; 1 + i\sqrt{3},\; 1 - i\sqrt{3}\)

34).

Let \(G\) be a group. A subgroup \(H\) of \(G\) is called characteristic if \(\varphi(H) \subset H\) for all automorphisms \(\varphi\) of \(G\). Pick out the true statement(s):

1. Every characteristic subgroup is normal.
2. Every normal subgroup is characteristic.
3. If \(N\) is a normal subgroup of a group \(G\), and \(M\) is a characteristic subgroup of \(N\), then \(M\) is a normal subgroup of \(G\).

Answer: (a), (c)

35).

Let \(G\) be a group and let \(H\) and \(K\) be subgroups of \(G\). The commutator subgroup \((H,K)\) is defined as the smallest subgroup containing all elements of the form \(hkh^{-1}k^{-1}\), where \(h \in H\) and \(k \in K\). Pick out the true statement(s):

1. If \(H\) and \(K\) are normal subgroups, then \((H,K)\) is a normal subgroup.
2. If \(H\) and \(K\) are characteristic subgroups, then \((H,K)\) is a characteristic subgroup.
3. \((G,G)\) is normal in \(G\) and \(G/(G,G)\) is abelian.

Answer: (a), (b), (c)

36).

Write the following permutation as a product of disjoint cycles:

\[ \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 6 & 5 & 4 & 3 & 1 & 2 \end{pmatrix} \]

Answer: \((1\,6\,2\,5)(3\,4)\)

37).

Pick out the true statement(s):

1. The set of all \(2 \times 2\) matrices with rational entries (with the usual operations of matrix addition and matrix multiplication) is a ring which has no non-trivial ideals.
2. Let \(R = C[0,1]\) be considered as a ring with the usual operations of pointwise addition and pointwise multiplication. Let \[ I = \{ f : [0,1] \to \mathbb{R} \mid f(1/2) = 0 \}. \] Then \(I\) is a maximal ideal.
3. Let \(R\) be a commutative ring and let \(P\) be a prime ideal of \(R\). Then \(R/P\) is an integral domain.

Answer: (a), (b), (c)

38).

What is the degree of the following numbers over \(\mathbb{Q}\)?

1. \(\sqrt{2} + \sqrt{3}\)
2. \(\sqrt{2} \sqrt{3}\)

Answer: (a) \(4\), (b) \(2\)

39).

Solve:

\[ x^4 - 3x^3 + 4x^2 - 3x + 1 = 0 \]

\[ 1,\ 1,\ \frac{1 + i\sqrt{3}}{2},\ \frac{1 - i\sqrt{3}}{2} \]

40).

Pick out the true statements:

1. Let \(H\) and \(K\) be subgroups of a group \(G\). For \(g \in G\), define the double coset \(HgK = \{h g k \mid h \in H, k \in K\}\). Then, if \(H\) is normal, we have \(HgH = gH\) for all \(g \in G\).
2. Let \(GL(n, \mathbb{C})\) be the group of all \(n \times n\) invertible matrices with complex entries. The set of all \(n \times n\) invertible upper triangular matrices is a normal subgroup.
3. Let \(M_n(\mathbb{R})\) denote the set of all \(n \times n\) matrices with real entries (with \(\mathbb{R}^n\) and usual topology), and let \(GL(n, \mathbb{R})\) denote the group of invertible matrices. Let \(G\) be a subgroup of \(GL(n, \mathbb{R})\). Define \( H = \{A \in G \mid \exists \varphi : [0,1] \to G \text{ continuous}\) \(, \varphi(0) = A,\ \varphi(1) = I \}.\)
   Then, \(H\) is a normal subgroup of \(G\).

Answer: (a), (c)

41).

How many (non-isomorphic) groups of order \(15\) are there?

Answer: \(1\)

42).

Pick out the true statements:

1. Let \(R\) be a commutative ring with identity. Let \(M\) be an ideal such that every element of \(R\) not in \(M\) is a unit. Then \(R/M\) is a field.
2. Let \(R\) be as above and let \(M\) be an ideal such that \(R/M\) is an integral domain. Then \(M\) is a prime ideal.
3. Let \(R = C[0,1]\) be the ring of real-valued continuous functions on \([0,1]\) with respect to pointwise addition and pointwise multiplication. Let \[ M = \{ f \in R \mid f(0) = f(1) = 0 \}. \] Then \(M\) is a maximal ideal.

Answer: (a), (b)

43).

Write down all the possible values for the degree of an irreducible polynomial in \(\mathbb{R}[x]\).

Answer: \(1, 2\)

44).

What is the quotient space \(M(n;\mathbb{R}) / T(n;\mathbb{R})\) isomorphic to?

Answer: \(\mathbb{R}\) (set of real numbers)

45).

Which of the following are subgroups of \(GL_3(\mathbb{C})\)?

(a)\; \[ H = \{ A \in M_3(\mathbb{C}) \mid \det(A) = 2^l,\ l \in \mathbb{Z} \}. \]

(b)\; \[ H = \left\{ \begin{pmatrix} 1 & \alpha & \beta \\ 0 & 1 & \gamma \\ 0 & 0 & 1 \end{pmatrix} \middle| \alpha,\beta,\gamma \in \mathbb{C} \right\}. \]

(c)\; \[ H = \left\{ \begin{pmatrix} 1 & 0 & \alpha \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \middle| \alpha \in \mathbb{C} \right\}. \]

Answer: (a), (b), (c)

46).

Let \(S_7\) denote the symmetric group of all permutations of the symbols \(\{1,2,3,4,5,6,7\}\). Pick out the true statements:

1. \(S_7\) has an element of order \(10\).
2. \(S_7\) has an element of order \(15\).
3. The order of any element of \(S_7\) is at most \(12\).

Answer: (a), (c)

47).

Let \(C(\mathbb{R})\) denote the ring of all continuous real-valued functions on \(\mathbb{R}\), with the operations of pointwise addition and pointwise multiplication. Which of the following form an ideal in this ring?

1. The set of all \(C^\infty\) functions with compact support.
2. The set of all continuous functions with compact support.
3. The set of all continuous functions which vanish at infinity, i.e. functions \(f\) such that \(\lim_{|x|\to\infty} f(x) = 0\).

Answer: (b)

48).

Find the number of non-zero elements in the field \(\mathbb{Z}_p\), where \(p\) is an odd prime number, which are squares, i.e. of the form \(m^2,\; m \in \mathbb{Z}_p,\; m \ne 0.\)

Answer: \(\frac{p-1}{2}\)

49).

Find the inverse in \(\mathbb{Z}_5\) of the following matrix:

\[ \begin{pmatrix} 1 & 2 & 0 \\ 0 & 2 & 4 \\ 0 & 0 & 3 \end{pmatrix} \]

\[ \begin{pmatrix} 1 & 4 & 3 \\ 0 & 3 & 1 \\ 0 & 0 & 2 \end{pmatrix} \]

50).

Find the number of elements of order two in the symmetric group \(S_4\) of all permutations of the four symbols \(\{1,2,3,4\}\).

Answer: \(9\)

51).

Let \(G\) be the group of all invertible \(2 \times 2\) upper triangular matrices (under matrix multiplication). Pick out the normal subgroups of \(G\) from the following:

1. \(H = \{A \in G : a_{12} = 0 \}\)
2. \(H = \{A \in G : a_{11} = 1 \}\)
3. \(H = \{A \in G : a_{11} = a_{22} \}\)
\[ \text{where}\; A \;= \; \begin{pmatrix} a_{11} & a_{12} \\ 0 & a_{22} \end{pmatrix} \]

Answer: (b), (c)

52).

Let \(G = GL_n(\mathbb{R})\) and let \(H\) be the (normal) subgroup of all matrices with positive determinant. Identify the quotient group \(G/H\).

Answer: \(\{1, -1\}\) multiplicative group

53).

Which of the following rings are integral domains?

1. \(\mathbb{R}[x]\), the ring of all polynomials in one variable with real coefficients.
2. \(M_n(\mathbb{R})\).
3. The ring of complex analytic functions defined on the unit disc of the complex plane (with pointwise addition and multiplication as the ring operations).

Answer: (a), (c)

54).

Let \(G\) be a finite group of order \(n \ge 2\). Which of the following statements are true?

1. There always exists an injective homomorphism from \(G\) into \(S_n\).
2. There always exists an injective homomorphism from \(G\) into \(S_m\) for some \(m < n\).
3. There always exists an injective homomorphism from \(G\) into \(GL_n(\mathbb{R})\).

Answer: (a), (c)

55).

Let \(\mathbb{C}^*\) denote the multiplicative group of non-zero complex numbers and let \(P\) denote the subgroup of positive real numbers. Identify the quotient group \(\mathbb{C}^* / P\).

Answer: multiplicative group of complex numbers with modulus one

56).

Given a finite group and a prime \(p\) which divides its order, let \(N(p)\) denote the number of \(p\)-Sylow subgroups of \(G\). If \(G\) is a group of order \(21\), what are the possible values of \(N(3)\) and \(N(7)\)?

Answer: \(N(3)=1 \text{ or } 7\) and \(N(7)=1\)

57).

Solve the following equation, given that its roots are in arithmetic progression:

\[ x^3 - 9x^2 + 28x - 30 = 0 \]

Answer: \(-3i,\; -3 + i,\; 3\)

58).

Which of the following statements are true?

1. Every group of order \(51\) is cyclic.
2. Every group of order \(151\) is cyclic.
3. Every group of order \(505\) is cyclic.

Answer: (a), (b)

59).

Let \(G\) be the multiplicative group of non-zero complex numbers. Consider the group homomorphism \(\varphi : G \to G\) given by \(\varphi(z) = z^4\).

1. Identify \(H\), the kernel of \(\varphi\).
2. Identify (up to isomorphism) the quotient space \(G/H\).

(a) \(H = \{1, -1, i, -i\}\)
(b) \(G\)

60).

How many elements of order \(7\) are there in a group of order \(28\)?

\(6\)

61).

Which of the following equations can occur as the class equation of a group of order \(10\)?

1. \(10 = 1 + 1 + 1 + 2 + 5\)
2. \(10 = 1 + 2 + 3 + 4\)
3. \(10 = 1 + 1 + \cdots + 1\) (10 times)

(c)

62).

With the usual notations, compute \(aba^{-1}\) in \(S_5\) and express it as the product of disjoint cycles, where

\[ a = (1\;2\;3)(4\;5), \quad b = (2\;3)(1\;4) \]

\((13)(25)(4)\)

63).

Consider the following permutation:

\[ \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 7 & 4 & 10 & 6 & 2 & 9 & 8 & 1 & 5 & 3 \end{pmatrix} \]

a. Is this an odd or an even permutation?
b. What is its order in \(S_{10}\)?

(a) odd
(b) 30

64).

Which of the following statements are true?

1. Let \(G\) be a group of order \(99\) and let \(H\) be a subgroup of order \(11\). Then \(H\) is normal in \(G\).
2. Let \(H\) be the subgroup of \(S_3\) consisting of \(\{e, a\}\) where \(a = (1\;2)\). Then \(H\) is normal in \(S_3\).
3. Let \(G\) be a finite group and \(H\) a subgroup of \(G\). Define \[ W = \bigcap_{g \in G} gHg^{-1} \] Then \(W\) is a normal subgroup of \(G\).

(a), (c)

65).

Consider the ring \(C[0,1]\) with the operations of pointwise addition and pointwise multiplication. Give an example of an ideal in this ring which is not a maximal ideal.

\(\{ f \in C[0,1] \mid f(0) = f(1) = 0 \}\)

66).

Compute the (multiplicative) inverse of \(4x + 3\) in the field \(\mathbb{Z}_{11}[x]/(x^2 + 1)\).

\(6x + 1\)

67).

Let \(G\) be a group. Which of the following statements are true?

1. Let \(H\) and \(K\) be subgroups of \(G\) of orders \(3\) and \(5\) respectively. Then \(H \cap K = \{e\}\), where \(e\) is the identity element of \(G\).
2. If \(G\) is an abelian group of odd order, then \(\varphi(x) = x^2\) is an automorphism of \(G\).
3. If \(G\) has exactly one element of order \(2\), then this element belongs to the centre of \(G\).

(a), (b), (c)

68).

Let \(n \in \mathbb{N},\, n \ge 2\). Which of the following statements are true?

1. Any finite group \(G\) of order \(n\) is isomorphic to a subgroup of \(GL_n(\mathbb{R})\).
2. The group \(\mathbb{Z}_n\) is isomorphic to a subgroup of \(GL_2(\mathbb{R})\).
3. The group \(\mathbb{Z}_{12}\) is isomorphic to a subgroup of \(S_7\).

(a), (b), (c)

69).

Which of the following statements are true?

1. The matrices \[ \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \text{ and } \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \] are conjugate in \(GL_2(\mathbb{R})\).
2. The matrices \[ \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \text{ and } \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \] are conjugate in \(SL_2(\mathbb{R})\).
3. The matrices \[ \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix} \text{ and } \begin{pmatrix} 1 & 3 \\ 0 & 2 \end{pmatrix} \] are conjugate in \(GL_2(\mathbb{R})\).

(a), (c)

70).

Let \(p\) be an odd prime. Find the number of non-zero squares in \(\mathbb{F}_p\).

\(p(p-1)/2\)

71).

Find a generator of \(\mathbb{F}_7^{\times}\), the multiplicative group of non-zero elements of \(\mathbb{F}_7\).

\(5\) or \(3\)

72).

Find the number of elements conjugate to \((1\;2\;3\;4\;5\;6\;7)\) in \(S_7\).

\(6!\)

73).

What is the order of a \(2\)-Sylow subgroup in \(\mathrm{GL}_3(\mathbb{F}_3)\)?

\(2^7\)

74).

Let \(H\) be the subgroup generated by \((1\;2)\) in \(S_3\). Compute the normalizer, \(N(H)\), of \(H\).

\(N(H) = H\)

75).

Let \(G\) be a group. Which of the following statements are true?

1. The normalizer of a subgroup of \(G\) is a normal subgroup of \(G\).
2. The centre of \(G\) is a normal subgroup of \(G\).
3. If \(H\) is a normal subgroup of \(G\) and is of order \(2\), then \(H\) is contained in the centre of \(G\).

(b), (c)

76).

Which of the following are prime ideals in the ring \(C[0,1]\)?

1. \(J = \{ f \in C[0,1] \mid f(x) = 0 \text{ for all } \tfrac{1}{3} \le x \le \tfrac{2}{3} \}\).
2. \(J = \{ f \in C[0,1] \mid f(\tfrac{1}{3}) = f(\tfrac{2}{3}) = 0 \}\).
3. \(J = \{ f \in C[0,1] \mid f(\tfrac{1}{3}) = 0 \}\).

(c)

77).

Which of the following statements are true?

1. If \(G\) is a finite group, then there exists \(n \in \mathbb{N}\) such that \(G\) is isomorphic to a subgroup of \(GL_n(\mathbb{R})\).
2. There exists an infinite group \(G\) such that every element, other than the identity element, is of order \(2\).
3. The group \(GL_2(\mathbb{R})\) contains a cyclic subgroup of order \(5\).

(b), (c)

78).

Let \(p\) be a prime number. Let \(n \in \mathbb{N}, n > 1\). What is the order of a \(p\)-Sylow subgroup of \(GL_n(\mathbb{F}_p)\)?

\(\displaystyle \frac{p^{n(n-1)/2}}{}\)

79).

Give an example of a \(5\)-Sylow subgroup of \(GL_3(\mathbb{F}_5)\).

\[ \begin{pmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{pmatrix} \]

80).

What is the number of elements of order \(2\) in \(S_4\)?

\(9\)

81).

Let \(G\) be a group of order \(10\). Which of the following could be the class equation of \(G\)?

1. \(10 = 1 + 1 + \cdots + 1\) (10 times)
2. \(10 = 1 + 2 + 3 + 4\)
3. \(10 = 1 + 1 + 1 + 2 + 5\)

(a)

82).

Find the number of irreducible monic polynomials of degree \(2\) in \(\mathbb{F}_p\), where \(p\) is a prime number.

\( \frac{p(p-1)}{2} \)

83).

The value of \(\alpha\) for which \(G = \{\alpha, 1, 3, 9, 19, 27\}\) is a cyclic group under multiplication modulo \(56\) is:

1. 5
2. 15
3. 25
4. 35

84).

Consider \(\mathbb{Z}_{24}\) as the additive group modulo \(24\). Then the number of elements of order \(8\) in the group \(\mathbb{Z}_{24}\) is:

1. 1
2. 2
3. 3
4. 4

85).

Let \(U(n)\) be the set of all positive integers less than \(n\) and relatively prime to \(n\). Then \(U(n)\) is a group under multiplication modulo \(n\). For \(n = 248\), the number of elements in \(U(n)\) is:

1. 60
2. 120
3. 180
4. 240

86).

Let \(\mathbb{R}[x]\) be the polynomial ring in \(x\) with real coefficients and let \(I = (x^2 + 1)\) be the ideal generated by the polynomial \(x^2 + 1\) in \(\mathbb{R}[x]\). Then:

1. \(I\) is a maximal ideal.
2. \(I\) is a prime ideal but NOT a maximal ideal.
3. \(I\) is NOT a prime ideal.
4. \(\mathbb{R}[x]/I\) has zero divisors.

87).

Consider \(\mathbb{Z}_5\) and \(\mathbb{Z}_{20}\) as rings modulo \(5\) and \(20\), respectively. Then the number of homomorphisms \(\varphi : \mathbb{Z}_5 \to \mathbb{Z}_{20}\) is:

1. 1
2. 2
3. 4
4. 5

88).

Let \(\mathbb{Q}\) be the field of rational numbers and consider \(\mathbb{Z}_2\) as a field modulo \(2\). Let \[ f(x) = x^3 - 9x^2 + 9x + 3. \] Then \(f(x)\) is:

1. irreducible over \(\mathbb{Q}\) but reducible over \(\mathbb{Z}_2\).
2. irreducible over both \(\mathbb{Q}\) and \(\mathbb{Z}_2\).
3. reducible over \(\mathbb{Q}\) but irreducible over \(\mathbb{Z}_2\).
4. reducible over both \(\mathbb{Q}\) and \(\mathbb{Z}_2\).

89).

Consider \(\mathbb{Z}_5\) as a field modulo \(5\) and let \[ f(x) = x^5 + 4x^4 + 4x^3 + 4x^2 + x + 1. \] Then the zeros of \(f(x)\) over \(\mathbb{Z}_5\) are \(1\) and \(3\) with respective multiplicity:

1. 1 and 4
2. 2 and 3
3. 2 and 2
4. 1 and 2

90).

Let \(G = \mathbb{R} \setminus \{0\}\) and \(H = \{-1, 1\}\) be groups under multiplication. Then the map \(\varphi : G \to H\) defined by \(\varphi(x) = \frac{x}{|x|}\) is:

1. not a homomorphism
2. a one-one homomorphism, which is not onto
3. an onto homomorphism, which is not one-one
4. an isomorphism

91).

The number of maximal ideals in \(\mathbb{Z}_{27}\) is:

1. 0
2. 1
3. 2
4. 3

92).

Let \(G\) be a group of order \(45\). Let \(H\) be a \(3\)-Sylow subgroup of \(G\) and \(K\) be a \(5\)-Sylow subgroup of \(G\). Then:

1. both \(H\) and \(K\) are normal in \(G\)
2. \(H\) is normal in \(G\) but \(K\) is not normal in \(G\)
3. \(H\) is not normal in \(G\) but \(K\) is normal in \(G\)
4. both \(H\) and \(K\) are not normal in \(G\)

93).

The ring \(\mathbb{Z}[\sqrt{-11}]\) is:

1. a Euclidean Domain
2. a Principal Ideal Domain, but not a Euclidean Domain
3. a Unique Factorization Domain, but not a Principal Ideal Domain
4. not a Unique Factorization Domain

94).

Let \(R\) be a Principal Ideal Domain and let \(a, b\) be any two non-unit elements of \(R\). Then the ideal generated by \(a\) and \(b\) is also generated by:

1. \(a + b\)
2. \(ab\)
3. \(\gcd(a, b)\)
4. \(\mathrm{lcm}(a, b)\)

95).

Consider the action of \(S_4\), the symmetric group of order 4, on \(\mathbb{Z}[x_1, x_2, x_3, x_4]\) given by \[ \sigma \cdot p(x_1, x_2, x_3, x_4) = p(x_{\sigma(1)}, x_{\sigma(2)}, x_{\sigma(3)}, x_{\sigma(4)}). \] \(\text{for} \; \sigma \in S_{4}\)
Let \(H \subseteq S_4\) denote the cyclic subgroup generated by \((1423)\). Then the cardinality of the orbit \(\mathcal{O}_H(x_1x_3 + x_2x_4)\) of H on the polynomial \(x_1x_3 + x_2x_4\) is:

1. 1
2. 2
3. 3
4. 4

96).

The number of elements of a principal ideal domain can be

1. 15
2. 25
3. 35
4. 36

97).

Which one of the following ideals of the ring \(\mathbb{Z}[i]\) of Gaussian integers is NOT maximal?

1. (\(1+i\))
2. (\(1-i\))
3. (\(2+i\))
4. (\(3+i\))

98).

If \(Z(G)\) denotes the centre of a group \(G\), then the order of the quotient group \(G/Z(G)\) cannot be

1. 4
2. 6
3. 15
4. 25

99).

Let \(\text{Aut}(G)\) denote the group of automorphisms of a group \(G\). Which one of the following is NOT a cyclic group?

1. \(\text{Aut}(\mathbb{Z}_4)\)
2. \(\text{Aut}(\mathbb{Z}_6)\)
3. \(\text{Aut}(\mathbb{Z}_8)\)
4. \(\text{Aut}(\mathbb{Z}_{10})\)

100).

Which one of the following groups is simple?

1. \(S_3\)
2. \(\text{GL}(2, \mathbb{R})\)
3. \(\mathbb{Z}_2 \times \mathbb{Z}_2\)
4. \(A_5\)

D

101).

Consider the algebraic extension \(E = \mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})\) of \(\mathbb{Q}\). Then \([E:\mathbb{Q}]\) is

1. 3
2. 4
3. 7
4. 8

D

102).

Let \(G_1\) be an abelian group of order 6 and \(G_2 = S_3\). For \(j = 1,2\), Let \(P_j\) be the statement “\(G_j\) has a unique subgroup of order \(2\)” for \(j = 1,2\). Then

1. both \(P_1\) and \(P_2\) hold
2. neither \(P_1\) nor \(P_2\) holds
3. \(P_1\) holds but not \(P_2\)
4. \(P_2\) holds but not \(P_1\)

C

103).

Let \(G\) be the group of all symmetries of the square. Then the number of conjugate classes in \(G\) is

1. 4
2. 5
3. 6
4. 7

B

104). Consider the polynomial ring \(\mathbb{Q}[x]\). The ideal of \(\mathbb{Q}[x]\) generated by \(x^2 - 3\) is:

(A) maximal but not prime
(B) prime but not maximal
(C) both maximal and prime
(D) neither maximal nor prime

C

105). The number of irreducible quadratic polynomials over the field of two elements \( \mathbb{F}_2 \) is:

(A) 0
(B) 1
(C) 2
(D) 3

106). The number of elements in the conjugacy class of the 3-cycle \((2\ 3\ 4)\) in the symmetric group \(S_6\) is:

(A) 20
(B) 40
(C) 120
(D) 216

107). If \(\mathbb{Z}[i]\) is the ring of Gaussian integers, the quotient \(\mathbb{Z}[i]/(3 - i)\) is isomorphic to:

(A) \(\mathbb{Z}\)
(B) \(\mathbb{Z}/3\mathbb{Z}\)
(C) \(\mathbb{Z}/4\mathbb{Z}\)
(D) \(\mathbb{Z}/10\mathbb{Z}\)

108). For the rings \(L = \frac{\mathbb{R}[x]}{(x^2 - x + 1)},\ M = \frac{\mathbb{R}[x]}{(x^2 + x + 1)},\) \(\ N = \frac{\mathbb{R}[x]}{(x^2 + 2x + 1)}\), which one of the following is TRUE?

(A) \(L\) is isomorphic to \(M\); \(L\) is not isomorphic to \(N\); \(M\) is not isomorphic to \(N\)
(B) \(M\) is isomorphic to \(N\); \(M\) is not isomorphic to \(L\); \(N\) is not isomorphic to \(L\)
(C) \(L\) is isomorphic to \(M\); \(M\) is isomorphic to \(N\)
(D) \(L\) is not isomorphic to \(M\); \(L\) is not isomorphic to \(N\); \(M\) is not isomorphic to \(N\)

109). The order of the smallest possible non-trivial group containing elements \(x\) and \(y\) such that \(x^7 = y^2 = e\) and \(yx = x^4y\) is:

(A) 1
(B) 2
(C) 7
(D) 14

B

110). The number of 5-Sylow subgroup(s) in a group of order 45 is:

(A) 1
(B) 2
(C) 3
(D) 4

A

111). Let \(\omega = \cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\). Let \(M = \begin{pmatrix}0 & i \\ i & 0\end{pmatrix}\) and \(N = \begin{pmatrix}\omega & 0 \\ 0 & \omega^2\end{pmatrix}\) and \(G = \langle M, N \rangle\) be the group generated by the matrices under multiplication. Then:

(A) \(G/Z(G) \cong C_6\)
(B) \(G/Z(G) \cong S_3\)
(C) \(G/Z(G) \cong C_2\)
(D) \(G/Z(G) \cong C_4\)

B

112). Suppose that \(R\) is a unique factorization domain and that \(a, b \in R\) are distinct irreducible elements. Which of the following statements is TRUE?

(A) The ideal \((1 + a)\) is a prime ideal
(B) The ideal \((a + b)\) is a prime ideal
(C) The ideal \((1 + ab)\) is a prime ideal
(D) The ideal \((a)\) is not necessarily a maximal ideal

D

113). Which of the following groups has a proper subgroup that is NOT cyclic?

(A) \(\mathbb{Z}_{15} \times \mathbb{Z}_{27}\)
(B) \(S_3\)
(C) \((\mathbb{Z}, +)\)
(D) \((\mathbb{Q}, +)\)

D

114). The number of group homomorphisms from \(\mathbb{Z}_3\) to \(\mathbb{Z}_9\) is ______.

3

115). Let \(G\) be a group of order 231. The number of elements of order 11 in \(G\) is ______.

10

116). Which of the following is a field?

(A) \(\mathbb{C}[x]/(x^2+2)\)
(B) \(\mathbb{Z}[x]/(x^2+2)\)
(C) \(\mathbb{Q}[x]/(x^2-2)\)
(D) \(\mathbb{R}[x]/(x^2-2)\)

C

117). Which of the following groups contains a unique normal subgroup of order four?

(A) \(\mathbb{Z}_2 \oplus \mathbb{Z}_4\)
(B) The dihedral group \(D_4\) of order eight
(C) The quaternion group \(Q_8\)
(D) \(\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2\)

All to Marks

118). The number of non-isomorphic groups of order 10 is ______.

2

119). Let \(a,b,c,d\) be real numbers with \(a < c < d < b\). Consider the ring \(C[a,b]\) with pointwise addition and multiplication. If \( S = {\, f \in C[a,b] : f(x) = 0 \text{ for all } x \in\)
\( [c,d] },\) then:
(A) \(S\) is NOT an ideal of \(C[a,b]\)
(B) \(S\) is an ideal of \(C[a,b]\) but NOT a prime ideal of \(C[a,b]\)
(C) \(S\) is a prime ideal of \(C[a,b]\) but NOT a maximal ideal of \(C[a,b]\)
(D) \(S\) is a maximal ideal of \(C[a,b]\)

B

120). Let \(R\) be a ring. If \(R[x]\) is a principal ideal domain, then \(R\) is necessarily a:
(A) Unique Factorization Domain
(B) Principal Ideal Domain
(C) Euclidean Domain
(D) Field

D

121). Consider the group homomorphism \(\varphi : M_2(\mathbb{R}) \to \mathbb{R}\) given by \(\varphi(A) = \mathrm{trace}(A)\). The kernel of \(\varphi\) is isomorphic to which of the following groups?
(A) \(M_2(\mathbb{R})/\{A \in M_{2}(\mathbb{R}): \phi(A) = 0\}\)
(B) \(\mathbb{R}^2\)
(C) \(\mathbb{R}^3\)
(D) \(GL_2(\mathbb{R})\)

C

122). Let \(\mathbb{F}_{125}\) be the field of \(125\) elements. The number of non-zero elements \(\alpha \in \mathbb{F}_{125}\) such that \(\alpha^5 = \alpha\) is ______.

4

123). Let \(c \in \mathbb{Z}_3\) be such that \(\displaystyle \mathbb{Z}_3[x]/(x^3 + cx + 1)\) is a field. Then \(c\) is equal to ______.

2

124). Let \(G = \{ e, x, x^2, x^3, y, xy, x^2y, x^3y \}\) with \(o(x) = 4\), \(o(y) = 2\) and \(xy = yx^3\). Then the number of elements in the center of the group \(G\) is:
(A) 1
(B) 2
(C) 4
(D) 8

B

125). The number of ring homomorphisms from \(\mathbb{Z}_2 \times \mathbb{Z}_2\) to \(\mathbb{Z}_4\) is ______.

1

126).
Let \( p(x) = 9x^5 + 10x^3 + 5x + 15 \) and \( q(x) = x^3 - x^2 - x - 2 \) be two polynomials in \( \mathbb{Q}[x] \). Then, over \( \mathbb{Q} \):

(A) \( p(x) \) and \( q(x) \) are both irreducible
(B) \( p(x) \) is reducible but \( q(x) \) is irreducible
(C) \( p(x) \) is irreducible but \( q(x) \) is reducible
(D) \( p(x) \) and \( q(x) \) are both reducible

C

127).
(P): If \( H \) is a normal subgroup of order 4 of the symmetric group \( S_4 \), then \( S_4/H \) is abelian.
(Q): If \( Q = \{\pm 1, \pm i, \pm j, \pm k\} \) is the quaternion group, then \( Q / \{ -1, 1 \} \) is abelian.

Which of the above statements hold TRUE?

(A) Both P and Q
(B) Only P
(C) Only Q
(D) Neither P nor Q

C

128).
Let \( F \) be a field of order 32. Then the number of non-zero solutions \( (a,b) \in F \times F \) of the equation \( x^2 + xy + y^2 = 0 \) is equal to __________.

-0.1 : 0.1

129).
The remainder when \( 98! \) is divided by \( 101 \) is equal to __________.

49.5 to 50.1

130).
Let \( G \) be a group whose presentation is

\[ G = \langle x,y \mid x^5 = y^2 = e,\quad x^2y = yx \rangle. \]
Then \( G \) is isomorphic to:

(A) \( \mathbb{Z}_5 \)
(B) \( \mathbb{Z}_{10} \)
(C) \( \mathbb{Z}_2 \)
(D) \( \mathbb{Z}_{30} \)

C

131). Let \(F_1\) and \(F_2\) be subfields of a finite field \(F\) consisting of \(2^9\) and \(2^6\) elements, respectively. Then the total number of elements in \(F_1 \cap F_2\) equals ______.

8

132). Let \(S_9\) be the group of all permutations of the set \(\{1,2,3,4,5,6,7,8,9\}\). Then the total number of elements of \(S_9\) that commute with \(\tau = (1\ 2\ 3)(4\ 5\ 6\ 7)\) in \(S_9\) equals ______.

24

133). Let \(\mathbb{Q}[x]\) be the ring of polynomials over \(\mathbb{Q}\). Then the total number of maximal ideals in the quotient ring \(\mathbb{Q}[x]/(x^4 - 1)\) equals ______.

3

134). In the permutation group \(S_6\), the number of elements of order \(8\) is:
(A) 0    (B) 1    (C) 2    (D) 4

A

135). Let \(R\) be a commutative ring with \(1\) (unity) which is not a field. Let \(I \subset R\) be a proper ideal such that every element of \(R\) not in \(I\) is invertible in \(R\). Then the number of maximal ideals of \(R\) is:
(A) 1    (B) 2    (C) 3    (D) infinite

A

136). Let \(F\) be a field with \(7^6\) elements and let \(K\) be a subfield of \(F\) with \(49\) elements. Then the dimension of \(F\) as a vector space over \(K\) is ______.

3

137). Consider the polynomial \(p(X) = X^4 + 4\) in the ring \(\mathbb{Q}[X]\) of polynomials in the variable \(X\) with coefficients in the field \(\mathbb{Q}\) of rational numbers. Then:
(A) the set of zeros of \(p(X)\) in \(\mathbb{C}\) forms a group under multiplication
(B) \(p(X)\) is reducible in the ring \(\mathbb{Q}[X]\)
(C) the splitting field of \(p(X)\) has degree \(3\) over \(\mathbb{Q}\)
(D) the splitting field of \(p(X)\) has degree \(4\) over \(\mathbb{Q}\)

B

138). Which one of the following statements is true?
(A) Every group of order \(12\) has a non-trivial proper normal subgroup
(B) Some group of order \(12\) does not have a non-trivial proper normal subgroup
(C) Every group of order \(12\) has a subgroup of order \(6\)
(D) Every group of order \(12\) has an element of order \(12\)

A

139). For an odd prime \(p\), consider the ring \(\mathbb{Z}[\sqrt{-p}] = \{a + b\sqrt{-p} : a, b \in \mathbb{Z}\} \subseteq \mathbb{C}\). Then the element \(2\) in \(\mathbb{Z}[\sqrt{-p}]\) is
(A) a unit (B) a square (C) a prime (D) irreducible

D

140). Consider the following statements: I. The set \(\mathbb{Q} \times \mathbb{Z}\) is uncountable.
II. The set \(\{f : f \text{ is a function from } \mathbb{N} \to \{0,1\}\}\) is uncountable.
III. The set \(\{\sqrt{p} : p \text{ is a prime number}\}\) is uncountable.
IV. For any infinite set, there exists a bijection from the set to one of its proper subsets.

Which of the above statements are TRUE?
(A) I and IV only    (B) II and IV only
(C) II and III only    (D) I, II and IV only

B

141). Let \(\omega\) be a primitive complex cube root of unity and \(i = \sqrt{-1}\). Then the degree of the field extension \(\mathbb{Q}(i,\sqrt{3},\omega)\) over \(\mathbb{Q}\) is ______.

4

142). Consider the following statements:
I. If \(\mathbb{Q}\) denotes the additive group of rational numbers and \(f : \mathbb{Q} \to \mathbb{Q}\) is a non-trivial homomorphism, then \(f\) is an isomorphism.
II. Any quotient group of a cyclic group is cyclic.
III. If every subgroup of a group \(G\) is a normal subgroup, then \(G\) is abelian.
IV. Every group of order \(33\) is cyclic.

Which of the above statements are TRUE?
(A) II and IV only    (B) II and III only
(C) I, II and IV only    (D) I, III and IV only

C

143). Consider the following statements:
I. The ring \(\mathbb{Z}[\sqrt{-1}]\) is a unique factorization domain.
II. The ring \(\mathbb{Z}[\sqrt{-5}]\) is a principal ideal domain.
III. In the polynomial ring \(\mathbb{Z}_2[x]\), the ideal generated by \(x^3 + x + 1\) is a maximal ideal.
IV. In the polynomial ring \(\mathbb{Z}_7[x]\), the ideal generated by \(x^6 + 1\) is a prime ideal.

Which of the above statements are TRUE?
(A) I, II and III only    (B) I and III only
(C) I and IV only    (D) II, III and IV only

B

144). The number of elements of order \(15\) in the additive group \(\mathbb{Z}_{60} \times \mathbb{Z}_{50}\) is ______.
\(\mathbb{Z}_n \;\) denotes the group of integers modulo n, under the operation of addition modulo n, for any positive integer n

48

145). Let \(\mathbb{Z}_{125}\) be the ring of integers modulo \(125\) under the operations of addition modulo \(125\) and multiplication modulo \(125\). If \(m\) is the number of maximal ideals of \(\mathbb{Z}_{125}\) and \(n\) is the number of non-units of \(\mathbb{Z}_{125}\), then \(m + n\) is equal to ______.

26

Q146).
Let G be a non-abelian group of order 125. Then the total number of elements in \( Z(G)=\{x \in G : gx = xg \text{ for all } g \in G \} \) equals ______.

5

Q147).
Let \( R = \mathbb{Z} \times \mathbb{Z} \times \mathbb{Z} \) and \( I = \mathbb{Z} \times \mathbb{Z} \times \{0\} \). Then which of the following statement is correct?

(A) \( I \) is a maximal ideal but not a prime ideal of \( R \).
(B) \( I \) is a prime ideal but not a maximal ideal of \( R \).
(C) \( I \) is both maximal ideal as well as a prime ideal of \( R \).
(D) \( I \) is neither a maximal ideal nor a prime ideal of \( R \).

B

Q.148).
Let \( G \) be a non-cyclic group of order \(57\). Then the number of elements of order \(3\) in \(G\) is ______.

38

Q.149).
Let \( K = \mathbb{Q}\!\left(\sqrt{3 + 2\sqrt{2}},\, \omega \right) \), where \( \omega \) is a primitive cube root of unity. Then the degree of extension of \(K\) over \(\mathbb{Q}\) is ______.

4

Q.150).
Let \( I \) and \( J \) be the ideals generated by \(\{5, \sqrt{10}\}\) and \(\{4, \sqrt{10}\}\) in the ring \(\mathbb{Z}[\sqrt{10}] = \{a + b\sqrt{10} \mid a,b \in \mathbb{Z}\}\), respectively. Then

(A) both \(I\) and \(J\) are maximal ideals
(B) \(I\) is a maximal ideal but \(J\) is not a prime ideal
(C) \(I\) is not a maximal ideal but \(J\) is a prime ideal
(D) neither \(I\) nor \(J\) is a maximal ideal

B

Q.151).
Let \(\mathbb{Z}_{225}\) be the ring of integers modulo \(225\). If \(x\) is the number of prime ideals and \(y\) is the number of nontrivial units in \(\mathbb{Z}_{225}\), then \(x + y\) is equal to ______.

121

Q.152).
The number of cyclic subgroups of the quaternion group \[ Q_8 = \langle a,b \mid a^4 = 1,\ a^2 = b^2,\ ba = a^3 b \rangle \] is ______.

5

Q.153).
The number of elements of order \(3\) in the symmetric group \(S_6\) is ______.

80

Q.154).
Let \(F\) be the field with \(4096\) elements. The number of proper subfields of \(F\) is ______.

5

Q.155).
Let \( R = \mathbb{Z} \times \mathbb{Z} \times \mathbb{Z} \) and \( I = \mathbb{Z} \times \mathbb{Z} \times \{0\} \). Then which of the following statement is correct?

(A) \(I\) is a maximal ideal but not a prime ideal of \(R\).
(B) \(I\) is a prime ideal but not a maximal ideal of \(R\).
(C) \(I\) is both maximal ideal as well as a prime ideal of \(R\).
(D) \(I\) is neither a maximal ideal nor a prime ideal of \(R\).

Q.156).
The number of \(5\)-Sylow subgroups in the symmetric group \(S_5\) of degree \(5\) is ______.

6

Q.157).
Let \( I \) be the ideal generated by \(x^2 + x + 1\) in the polynomial ring \( R = \mathbb{Z}_3[x] \), where \( \mathbb{Z}_3 \) denotes the ring of integers modulo \(3\). Then the number of units in the quotient ring \( R/I \) is ______.

6

Q.158).
Let \( G \) be a group of order \(5^4\) with center having \(5^2\) elements. Then the number of conjugacy classes in \(G\) is ______.

145

Q.159).
Let \(F\) be a finite field and \(F^\times\) be the group of all nonzero elements of \(F\) under multiplication. If \(F^\times\) has a subgroup of order \(17\), then the smallest possible order of the field \(F\) is ______.

103

Q.160).
Let \(\mathbb{Z}\) denote the ring of integers. Consider the subring \[ R = \{\, a + b\sqrt{-17} : a, b \in \mathbb{Z} \,\} \] of the field \(\mathbb{C}\) of complex numbers.

Consider the following statements:
\(P:\) \( 2 + \sqrt{-17} \) is an irreducible element.
\(Q:\) \( 2 + \sqrt{-17} \) is a prime element.

Then:
(A) both \(P\) and \(Q\) are TRUE
(B) \(P\) is TRUE and \(Q\) is FALSE
(C) \(P\) is FALSE and \(Q\) is TRUE
(D) both \(P\) and \(Q\) are FALSE

B

Q.161

The number of non isomorphic abelian groups of order \(2^3.3^3.5^4\) is ______.

30

Q.162

The number of subgroups of a cyclic group of order \(12\) is ______.

6

Q.163

Let \(\mathbb{R}[X]\) denote the ring of polynomials in \(X\) with real coefficients. Then, the quotient ring \(\mathbb{R}[X]/(X^{4}+4)\) is

(A) a field

(B) an integral domain, but not a field

(C) not an integral domain, but has \(0\) as the only nilpotent element

(D) a ring which contains non-zero nilpotent elements

C

Q.164

Consider the following conditions on two proper non-zero ideals \(J_{1}\) and \(J_{2}\) of a non-zero commutative ring \(R\).
\(P\colon\) For any \(r_{1},r_{2}\in R\), there exists a unique \(r\in R\) such that \(r-r_{1}\in J_{1}\) and \(r-r_{2}\in J_{2}\).
\(Q\colon J_{1}+J_{2}=R\).

Then, which of the following statements is TRUE?

(A) \(P\) implies \(Q\) but \(Q\) does not imply \(P\)

(B) \(Q\) implies \(P\) but \(P\) does not imply \(Q\)

(C) \(P\) implies \(Q\) and \(Q\) implies \(P\)

(D) \(P\) does not imply \(Q\) and \(Q\) does not imply \(P\)

A

Q.165

Let \(G\) be a group of order \(5^{4}\) with center having \(5^{2}\) elements. Then the number of conjugacy classes in \(G\) is ______.

D

Q.166

Let \(F\) be a finite field and \(F^{\times}\) be the group of all nonzero elements of \(F\) under multiplication. If \(F^{\times}\) has a subgroup of order \(17\), then the smallest possible order of the field \(F\) is ______.

Q.167. Let \(GL_2(\mathbb{C})\) denote the group of \(2 \times 2\) invertible complex matrices with usual matrix multiplication. For \(S,T \in GL_2(\mathbb{C})\), \(\langle S,T \rangle\) denotes the subgroup generated by \(S\) and \(T\). Let \[ S=\begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}\in GL_2(\mathbb{C}) \] and let \(G_1,G_2,G_3\) be three subgroups of \(GL_2(\mathbb{C})\) given by \[ G_1 = \langle S,T_1\rangle,\quad T_1 = \begin{bmatrix} i & 0\\ 0 & i\end{bmatrix}; \] \[ G_2 = \langle S,T_2\rangle,\quad T_2 = \begin{bmatrix} i & 0\\ 0 & -i\end{bmatrix}; \] \[ G_3 = \langle S,T_3\rangle,\quad T_3 = \begin{bmatrix} 0 & 1\\ 1 & 0\end{bmatrix}. \] Let \(Z(G_i)\) denote the center of \(G_i\) for \(i=1,2,3\). Which of the following statements is correct?

1. \(G_1\) is isomorphic to \(G_3\)
2. \(Z(G_1)\) is isomorphic to \(Z(G_2)\)
3. \(Z(G_3)=\left\{\begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}\right\}\)
4. \(Z(G_2)\) is isomorphic to \(Z(G_3)\)

D

Q.168. Let \(\sigma \in S_8\), where \(S_8\) is the permutation group on \(8\) elements. Suppose \(\sigma\) is the product of \(\sigma_1\) and \(\sigma_2\), where \(\sigma_1\) is a \(4\)-cycle and \(\sigma_2\) is a \(3\)-cycle in \(S_8\). If \(\sigma_1\) and \(\sigma_2\) are disjoint cycles, then the number of elements in \(S_8\) which are conjugate to \(\sigma\) is \(\underline{\hspace{3cm}}\).

3360

Q.169. Let \(G\) be an abelian group and \(\Phi : G \to (\mathbb{Z},+)\) be a surjective group homomorphism. Let \(1 = \Phi(a)\) for some \(a \in G\). Consider the following statements:

\(\mathbf{P}:\) For every \(g \in G\), there exists an \(n \in \mathbb{Z}\) such that \(g a^{n} \in \ker(\Phi)\).
\(\mathbf{Q}:\) Let \(e\) be the identity of \(G\) and \(\langle a \rangle\) be the subgroup generated by \(a\). Then \(G = \ker(\Phi)\langle a \rangle\) and \(\ker(\Phi)\cap \langle a \rangle = \{e\}\).

Which of the following statements is/are correct?

1. \(\mathbf{P}\) is TRUE
2. \(\mathbf{P}\) is FALSE
3. \(\mathbf{Q}\) is TRUE
4. \(\mathbf{Q}\) is FALSE

A, C

Q.170. Let \[ R = \{\,p(x) \in \mathbb{Q}[x] : p(0) \in \mathbb{Z}\,\}, \] where \(\mathbb{Q}\) denotes the set of rational numbers and \(\mathbb{Z}\) denotes the set of integers. For \(a \in R\), let \(\langle a \rangle\) denote the ideal generated by \(a\) in \(R\). Which of the following statements is/are correct?

1. If \(p(x)\) is an irreducible element in \(R\), then \(\langle p(x)\rangle\) is a prime ideal in \(R\).
2. \(R\) is a unique factorization domain.
3. \(\langle x\rangle\) is a prime ideal in \(R\).
4. \(R\) is NOT a principal ideal domain.

A

Q.171. Consider the rings \[ S_1 = \mathbb{Z}[x]/\langle 2,x^3\rangle \quad\text{and}\quad S_2 = \mathbb{Z}_2[x]/\langle x^2\rangle, \] where \(\langle 2,x^3\rangle\) denotes the ideal generated by \(\{2,x^3\}\) in \(\mathbb{Z}[x]\) and \(\langle x^2\rangle\) denotes the ideal generated by \(x^2\) in \(\mathbb{Z}_2[x]\). Which of the following statements is/are correct?

1. Every prime ideal of \(S_1\) is a maximal ideal.
2. \(S_2\) has exactly one maximal ideal.
3. Every element of \(S_1\) is either nilpotent or a unit.
4. There exists an element in \(S_2\) which is neither nilpotent nor a unit.

A, B, C

Q.172

Let \(K \subseteq \mathbb{C}\) be the field extension of \(\mathbb{Q}\) obtained by adjoining all the roots of the polynomial equation \((X^{2}-2)(X^{2}-3)=0\). The number of distinct fields \(F\) such that \(\mathbb{Q} \subseteq F \subseteq K\) is equal to ______ (answer in integer).

5

Q.173

The number of non-isomorphic finite groups with exactly 3 conjugacy classes is equal to ______ (answer in integer).

2

Q.174

Let \(\mathbb{F}_{3}\) be the field with exactly 3 elements. The number of elements in \(GL_{2}(\mathbb{F}_{3})\) is equal to ______ (answer in integer).

48

Q.175

Consider the following statements:
I. There exists a proper subgroup \(G\) of \((\mathbb{Q},+)\) such that \(\mathbb{Q}/G\) is a finite group.
II. There exists a subgroup \(G\) of \((\mathbb{Q},+)\) such that \(\mathbb{Q}/G\) is isomorphic to \((\mathbb{Z},+)\).
Which one of the following is correct?
(A) Both I and II are TRUE
(B) I is TRUE and II is FALSE
(C) I is FALSE and II is TRUE
(D) Both I and II are FALSE

D

Q.176

Let \(\mathbb{R}[X^{2},X^{3}]\) be the subring of \(\mathbb{R}[X]\) generated by \(X^{2}\) and \(X^{3}\). Consider the following statements:
I. The ring \(\mathbb{R}[X^{2},X^{3}]\) is a unique factorization domain.
II. The ring \(\mathbb{R}[X^{2},X^{3}]\) is a principal ideal domain.
Which one of the following is correct?
(A) Both I and II are TRUE
(B) I is TRUE and II is FALSE
(C) I is FALSE and II is TRUE
(D) Both I and II are FALSE

D

Q.177

Given a prime number \(p\), let \(n_{p}(G)\) denote the number of \(p\)-Sylow subgroups of a finite group \(G\). Which one of the following is TRUE for every group \(G\) of order \(2024\)?
(A) \(n_{11}(G)=1\) and \(n_{23}(G)=11\)
(B) \(n_{11}(G)\in\{1,23\}\) and \(n_{23}(G)=1\)
(C) \(n_{11}(G)=23\) and \(n_{23}(G)\in\{1,88\}\)
(D) \(n_{11}(G)=23\) and \(n_{23}(G)=11\)

B

Q.178

The number of group homomorphisms from \(\mathbb{Z}/4\mathbb{Z}\) to \(S_{4}\) is equal to ______ (answer in integer).

16

Q.179.

Let \(K \subseteq \mathbb{C}\) be the field extension of \(\mathbb{Q}\) obtained by adjoining all the roots of the polynomial equation \((X^2-2)(X^2-3)=0\). The number of distinct fields \(F\) such that \(\mathbb{Q} \subseteq F \subseteq K\) is equal to _______ (answer in integer).

16

Q.180.

Let \(G\) be a group with identity element \(e\), and let \(g,h\in G\) be such that the following hold:
(i) \(g\neq e,\; g^2=e\),
(ii) \(h\neq e,\; h^2\neq e,\) and \(ghg^{-1}=h^2\).
Then, the least positive integer \(n\) for which \(h^n=e\) is _______ (in integer).

3

Q.181.

Let \(E\subset F\) and \(F\subset K\) be field extensions which are not algebraic. Let \(\alpha\in K\) be algebraic over \(F\) and \(\alpha\notin F\). Let \(L\) be the subfield of \(K\) generated over \(E\) by the coefficients of the monic polynomial of minimal degree over \(F\) which has \(\alpha\) as a zero. Then, which of the following is/are TRUE?

(A) \(F(\alpha)\supset L(\alpha)\) is a finite extension if and only if \(F\supset L\) is a finite extension

(B) The dimension of \(L(\alpha)\) over \(L\) is greater than the dimension of \(F(\alpha)\) over \(F\)

(C) The dimension of \(L(\alpha)\) over \(L\) is smaller than the dimension of \(F(\alpha)\) over \(F\)

(D) \(F(\alpha)\supset L(\alpha)\) is an algebraic extension if and only if \(F\supset L\) is an algebraic extension

A, D

Q.182.

All rings considered below are assumed to be associative and commutative with \(1\neq 0\). Further, all ring homomorphisms map 1 to 1.
Consider the following statements about such a ring \(R\):
\(P_1:\; R\) is isomorphic to the product of two rings \(R_1\) and \(R_2\).
\(P_2:\; \exists\, r_1,r_2\in R\) such that \(r_1^2=r_1\neq 0\neq r_2=r_2^2,\; r_1r_2=0\) and \(r_1+r_2=1\).
\(P_3:\; \exists\) ideals \(I_1,I_2\subseteq R\) with \(R\neq I_1\neq (0)\neq I_2\neq R\) such that \(R=I_1+I_2\) and \(I_1\cap I_2=(0)\).
\(P_4:\; \exists\, a,b\in R\) with \(a\neq 0\neq b\) such that \(ab=0\).
Then, which of the following is/are TRUE?

(A) \(P_1 \Rightarrow P_2\)

(B) \(P_2 \Rightarrow P_3\)

(C) \(P_3 \Rightarrow P_4\)

(D) \(P_4 \Rightarrow P_1\)

A, B, C

Q.183.

Let \(K\) be an algebraically closed field containing a finite field \(F\). Let \(L\) be the subfield of \(K\) consisting of elements of \(K\) that are algebraic over \(F\). Consider the following statements:
S1: \(L\) is algebraically closed.
S2: \(L\) is infinite.
Then, which one of the following is correct?

(A) S1 is TRUE and S2 is FALSE

(B) S2 is TRUE and S1 is FALSE

(C) Both S1 and S2 are TRUE

(D) Neither S1 nor S2 is TRUE

C