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

The Correct Option is B, C

- (A) This statement is false because if \( G \) is non-abelian and the center contains more than one element, the quotient group \( G/Z(G) \) is also non-trivial and can contain more than one element. - (B) For any group \( G \) of order \( |G| \geq 2 \), there exists a non-trivial homomorphism from \( \mathbb{Z} \) (the additive group of integers) to \( G \). This follows from the fact that any group of order greater than 1 has a non-trivial homomorphism from \( \mathbb{Z} \). - (C) If \( |G| \geq 2 \) and \( G \) is non-abelian, the existence of a non-identity isomorphism from \( G \) to itself is guaranteed by group theory results, especially in non-abelian groups with order greater than 2. - (D) This statement is false. A group of order \( p^3 \), where \( p \) is a prime, is not necessarily abelian. In fact, there exist non-abelian groups of order \( p^3 \) (for example, the Heisenberg group). Thus, the correct answers are (B) and (C).