List of top Linear Algebra Questions on Groups

Consider
\(𝐺 = \left\{𝑚 + 𝑛\sqrt2\ ∶\ 𝑚, 𝑛 ∈ \Z\right\}\)
as a subgroup of the additive group ℝ.
Which of the following statements is/are TRUE ?

Which of the following statements is/are TRUE ?

The center Z(G) of a group G is defined as
z(G) = {x ∈ G ∶ xg = gx for all g ∈ G}.
Let |G| denote the order of G. Then, which of the following statements is/are TRUE for any group G ?

Which one of the following is TRUE for the symmetric group S₁₃ ?

From the additive group Q to which one of the following groups does there exist a non-trivial group homomorphism ?

Let G be a finite group. Then G is necessarily a cyclic group if the order of G is

Consider the following statements :
I. Every infinite group has infinitely many subgroups.
II. There are only finitely many non-isomorphic groups of a given finite order.
Then

How many group homomorphisms are there from \(\Z_2\) to S₅ ?

Let A ⊆ \(\Z\) with 0 ∈ A. For r, s ∈ \(\Z\), define
rA = {ra : a ∈ A},   rA + sA = {ra + sb : a, b ∈ A}.
Which of the following conditions imply that A is a subgroup of the additive group \(\Z\) ?

For σ ∈ S₈, let o(σ) denote the order of σ. Then max{o(σ) : σ ∈ S₈} is equal to __________.

For g ∈ \(\Z\), let \(\bar{g}\) ∈ \(\Z_8\) denote the residue class of g modulo 8. Consider the group \(\Z^×_8\) = {\(\bar{x}\) ∈ \(\Z_8\) : 1 ≤ x ≤ 7, gcd(x, 8) = 1} with respect to multiplication modulo 8. The number of group isomorphisms from \(\Z^×_8\) onto itself is equal to ________

The number of permutations in S₄ that have exactly two cycles in their cycle decompositions is equal to ___________.

Which one of the following groups has elements of order 1, 2, 3, 4, 5 but does not have an element of order greater than or equal to 6 ?

Consider the group G = {A ∈ M₂ (ℝ): AA^T = I₂} with respect to matrix multiplication. Let
Z(G) = {A ∈ G : AB = BA, for all B ∈ G}.
Then, the cardinality of Z(G) is

Let G be a group of order 39 such that it has exactly one subgroup of order 3 and exactly one subgroup of order 13. Then, which one of the following statements is TRUE ?

For a positive integer n, let U(n) = {\(\bar{r}\) ∈ ℤ_n ∶ gcd(r, n) = 1} be the group under multiplication modulo n. Then, which one of the following statements is TRUE ?

Let G be a finite group containing a non-identity element which is conjugate to its inverse. Then, which one of the following is TRUE ?

Let \( G \) be a group of order 2022. Let \( H \) and \( K \) be the subgroups of \( G \) of order 337 and 674, respectively. If \( H \cup K \) is also a subgroup of \( G \), then which one of the following is FALSE?

Consider the group \( (\mathbb{Q}, +) \) and its subgroup \( (\mathbb{Z}, +) \).
For the quotient group \( \mathbb{Q}/\mathbb{Z} \), which one of the following is FALSE?

For \(g\isin\Z\), let \(\bar{g}\isin\Z_{37}\) denote the residue class of g modulo 37. Consider the group U₃₇ = {\(\bar{g}\isin\Z_{37}:1\le g\le37\) with gcd(g, 37) = 1} with respect to multiplication modulo 37. Then which one of the following is FALSE?

The number of distinct subgroup of Z₉₉₉ is_____.

The number of elements of order 12 in the symmetric group S₇ is equal to____

Let σ be the permutation in the symmetric group S₅ given by
σ(1) = 2, σ(2) = 3, σ(3) = 1, σ(4) = 5, σ(5) = 4.
Define
N(σ) = {𝜏 ∈ S₅ ∶ σ ∘ 𝜏 = 𝜏 ∘ σ}.
Then the number of elements in N(σ) is equal to _________.

Let be a finite group of order at least two and let e denote the identity element of G. Let σ: G →g be a bijective group homomorphism that satisfies the following two conditions:
(i): if σ(g)=g for some gεG, then g=e, 
(ii) (σ o σ) (g)=g for all g σ G. 
The n which of the following is/are correct ?