CSIR NET Maths previous year question papers

Single choice · Groups

1. A

   p ∤ 1 + (p − 1)! for some odd prime p.

2. B

   p | $(1234)^{p-1}$ − 1 for all primes p > 700.

3. C

   There exist $a \in \mathbb{Z}$ and a prime p > 11 such that p ∤ aᵖ − a.

4. D

   p ∤ $(p^{2})!/(p!)^{2}$ for some odd prime p.

Single choice · Groups

1. A

   The number of pairs $(\sigma, x) \in G \times X$ such that $\sigma(x) = x$ is strictly greater than $\sum_{x}$ |S_G(x)|.

2. B

   The number (1/|G|$)\cdot\sum_{x}$ |S_G(x)| is always an integer.

3. C

   For all x ∈ X, S_G(x) is a normal subgroup of G.

4. D

   For all x, y ∈ X, S_G(x) is isomorphic to S_G(y).

Single choice · Groups

1. A

   Let G and G′ be finite groups such that S(G) = S(G′). Then G is isomorphic to G′.

2. B

   If S(G) = 4, then |G| = pᵐ for some prime number p and positive integer m.

3. C

   If S(G) = 5, then G is a cyclic group.

4. D

   For every positive integer n, there exists a finite group G such that S(G) = n.

Single choice · Groups

1. A

   There exists a finite group G such that |H_G| is even.

2. B

   There exists a finite group G such that |H_G| = 4n+1 for some n ≥ 3.

3. C

   For every finite group G, there exists a non-negative integer n such that |H_G| = 4n+1.

4. D

   For every finite group G, there exists a non-negative integer n such that |H_G| = 4n+3.

Multi-select · full marks only for the exact set · Groups

1. A

   G = GL$_{2}($𝔽_p) and H is a subgroup of order p.

2. B

   G = SL$_{2}($𝔽_p) and H is a subgroup of order p.

3. C

   p ≡ 3 (mod 4), G = GL$_{2}($𝔽_p)/SL$_{2}($𝔽_p) and H is a subgroup of order 2.

4. D

   p ≡ 1 (mod 4), G = GL$_{2}($𝔽_p)/SL$_{2}($𝔽_p) and H is a subgroup of order 2.

Multi-select · full marks only for the exact set · Groups

1. A

   If $G_{1}, G_{2}$ are two groups such that Aut$(G_{1})$ is isomorphic to Aut$(G_{2})$, then $G_{1}$ is isomorphic to $G_{2}$.

2. B

   If |G| = 2, then Aut(G × G) is abelian.

3. C

   If G is the group of complex numbers under addition, then Aut(G) is abelian.

4. D

   If G is finite, then Aut(G) is finite.

Multi-select · full marks only for the exact set · Groups

1. A

   If $G_{1}$ is normal in G, then $(G_{2}G_{1})/G_{1} \cong G_{2}/(G_{2} \cap G_{1})$.

2. B

   If $H_{1}$ and $H_{2}$ are normal subgroups of $G_{1}$ and $G_{2}$, respectively, then $(G_{1} \times G_{2})/(H_{1} \times H_{2}) \cong (G_{1}/H_{1}) \times (G_{2}/H_{2})$.

3. C

   If $G_{1}$ is normal in $G_{2}$ and $G_{2}$ is normal in G, then $G_{1}$ is normal in G.

4. D

   Every subgroup of prime index in G is normal.

Multi-select · full marks only for the exact set · Groups

1. A

   If there are no elements of order 4 in G, then G is abelian.

2. B

   If there are exactly two elements of order 4 in G, then G is abelian.

3. C

   If there are exactly six elements of order 4 in G, then G is abelian.

4. D

   There is an element of order 4 in G.

Multi-select · full marks only for the exact set · Groups

1. A

   If d is a positive integer that divides |G|, then G has a subgroup of order d.

2. B

   The map f : G×G → G given by f(a,b)=ab is not a group homomorphism.

3. C

   Suppose that for every positive integer d that divides |G|, there exists a subgroup of G of order d. Then G has at least three normal subgroups.

4. D

   |G| ≠ 16

Single choice · Groups

1. A

   For each $p < 100, \sum(n=1..\infty) (S_p/\varphi(p))^{n}$ converges.

2. B

   For each $100 \le p \le 200, \sum(n=1..\infty) (S_p/\varphi(p))^{n}$ diverges.

3. C

   For each $p > 200, \sum(n=1..\infty) (S_p/\varphi(p))^{n}$ converges.

4. D

   The element 4 mod 101 in $(\mathbb{Z}/101\mathbb{Z})$ˣ is a primitive root.

Single choice · Groups

1. A

   If G is abelian, then |X| = n.

2. B

   If |X| divides n, then G is abelian.

3. C

   |X| < n

4. D

   |X| divides n.

Single choice · Groups

1. A

   $10^{7}$

2. B

   10!/7!

3. C

   10!/3!

4. D

   $7^{10}$

Single choice · Groups

1. A

   If G(n) = 1, then n is prime.

2. B

   G(8) = 2

3. C

   If $\gcd(n, \varphi(n)) > 1$, then G(n) > 1. (Here $\varphi$ denotes the Euler $\varphi-$function.)

4. D

   $\lim \sup(n\to\infty) G(n) = 2$

Single choice · Groups

1. A

   The cyclic group of order 6.

2. B

   The symmetric group $S_{5}$.

3. C

   The alternating group $A_{5}$.

4. D

   The dihedral group with ten elements.

Single choice · Groups

1. A

   525

2. B

   35

3. C

   840

4. D

   315

Single choice · Groups

1. A

   30

2. B

   60

3. C

   45

4. D

   10

Multi-select · full marks only for the exact set · Groups

1. A

   If p divides |G|, then E(p) is a subgroup of G.

2. B

   For all p, E(p) is a subgroup of G.

3. C

   If E(p) is a subgroup of G, then p divides |G|.

4. D

   If G is abelian, then E(p) is a subgroup of G.

Multi-select · full marks only for the exact set · Groups

1. A

   If an abelian group G has property (P), every subgroup of G has property (P).

2. B

   If an abelian group G has property (P), then every element of G has finite order.

3. C

   The group $\mathbb{Z}$ has property (P).

4. D

   If an abelian group G has property (P), then no element has order $p^{2}$, where p is a prime number.

Multi-select · full marks only for the exact set · Groups

1. A

   Any group with p(p + 2) elements, where p and p + 2 are both prime numbers, has property (J).

2. B

   Any finite abelian group has property (J).

3. C

   The symmetric group $S_{3}$ has property (J).

4. D

   Any group with 77 elements has property (J).

Multi-select · full marks only for the exact set · Groups

1. A

   $\mathbb{Q}$ with ordinary addition

2. B

   $\mathbb{C}$ \ {0} with ordinary multiplication

3. C

   The cyclic group of order 5

4. D

   The symmetric group $S_{5}$

Multi-select · full marks only for the exact set · Groups

1. A

   3

2. B

   4

3. C

   5

4. D

   6

Single choice · Groups

1. A

   If |G| $= p^{6}$, then G has a subgroup of index $p^{2}$.

2. B

   If |G| $= p^{6}$, then G has at least five normal subgroups.

3. C

   Center of G can be infinite.

4. D

   There exists G with |G| $= p^{6}$ such that G has exactly six normal subgroups.

Single choice · Groups

1. A

   G is a union of proper subgroups.

2. B

   G is a union of proper subgroups if |G| has at least two distinct prime divisors.

3. C

   If G is abelian, then G is a union of proper subgroups.

4. D

   G is a union of proper subgroups if and only if G is not cyclic.

Multi-select · full marks only for the exact set · Groups

1. A

   G is an Abelian group.

2. B

   G is a cyclic group.

3. C

   G is a simple group.

4. D

   G is not a simple group.

Multi-select · full marks only for the exact set · Groups

1. A

   a is divisible by $3^{4}$.

2. B

   a is divisible by $2^{4}$.

3. C

   a is not divisible by 48.

4. D

   a is divisible by $3^{6}$.