Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 3 / 2D (Groups)

$Z(G)=\{z:zg=gz\text{ for every }g\in G\}$; it is closed under products and inverses, abelian, and invariant under conjugation. If $K=\{e,k\}$ is normal, conjugation fixes its unique nonidentity element $k$, so $k\in Z(G)$. Thus the statement is true.

True. If $G/Z(G)$ is generated by $gZ(G)$, write $x=g^a{z}_1$, $y=g^b{z}_2$ with central $z_i$; then $xy=yx$.

True. $S_3$ has trivial centre. A central permutation would commute with both $(12)$ and $(123)$, which only the identity does.