Question

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 ?

• A. If G is non-abelian and Z(G) contains more than one element, then the center of the quotient group G/Z(G) contains only one element
• B. If |G| ≥ 2, then there exists a non-trivial homomorphism from Z to G
• C. If |G| ≥ 2 and G is non-abelian, then there exists a non-identity isomorphism from G to itself
• D. If |G| = p3 , where 𝑝 is a prime number, then 𝐺 is necessarily abelian

Answer 1

Answer: (B), (C)

If |G| ≥ 2, then there exists a non-trivial homomorphism from Z to G

Answer 2

The correct option is (B) : If |G| ≥ 2, then there exists a non-trivial homomorphism from Z to G and (C) : If |G| ≥ 2 and G is non-abelian, then there exists a non-identity isomorphism from G to itself