$G$ is a finite group, prove that $G$ is abelian when $G/C(G)$ is cyclic.

Question

Is it true that when $G/C(G)$ is cyclic, then $G$ is abelian?

The original question is

Prove that if $G$ is a finite non-abelian group, then $4|C(G)|\le G$.
Proof from the answer book: suppose that $1<G/C(G)<4$, then $G/C(G)$ is of order 2 or 3 and therefore cyclic. Hence $G$ would be abelian, a contradiction.

As is said in the title, I have no idea how the implication in the last sentence goes...

https://proofwiki.org/wiki/Quotient_of_Group_by_Center_Cyclic_implies_Abelian — Cloudscape, Sep 21 '15 at 08:18
(Note that $Z(G)$ is another notation for the center $C(G)$ of $G$. You don't even need $G$ to be finite.) — Najib Idrissi, Sep 21 '15 at 08:34

score 4 · Accepted Answer · answered Sep 21 '15 at 08:19

Let $aC(G)$ be a coset which generates $G/C(G)$, so that $$ G=\bigsqcup_{n\geq 0}a^nC(G). $$ Consider any two elements $g,h\in G$. We can write them as $g=a^nx$ and $h=a^my$ where $x,y\in C(G)$. Then $gh=a^nxa^my=a^{n+m}xy=hg$, where we have commuted $x$ and $y$ past everything (because they belong to the center). Hence $G$ is abelian.

score 1 · Answer 2 · answered Sep 21 '15 at 08:26

1

A slight variation on pre-kidney's answer: Suppose $a$ is a cyclic generator of $G/C(G)$. Then for any $g\in G$ we have $g=a^nx$ where $x\in C(G)$, so $ag=aa^nx=a^nax=a^nxa=ga$, so $a\in C(G)$ and $G=C(G)$. Hence $G$ is abelian.

answered Sep 21 '15 at 08:26

Mario Carneiro

28,221

$G$ is a finite group, prove that $G$ is abelian when $G/C(G)$ is cyclic.

2 Answers2