If a group $G$ has a nontrivial centre, must every subgroup of index $3$ be normal?

$S_3$ yields an example of a group with a non-normal subgroup of index $3$, although it has a trivial centre. Moreover, for finite $G$, it's well-known that if $p$ is the smallest prime dividing $|G|$, then any subgroup of index $p$ is normal. Hence the answer to this question is "yes" if $G$ is a finite group of odd order divisible by $3$.

I'm considering dihedral groups as possible counterexamples, but haven't come up with anything.

  • 3
    $\begingroup$ Can’t you fancy up your permutation example into a counterexample by direct products? $\endgroup$ – Randall Dec 31 '18 at 0:25
  • $\begingroup$ Randall's correct; you can just take $S_3 \times C_2$. $\endgroup$ – Qiaochu Yuan Dec 31 '18 at 0:40
  • $\begingroup$ @Randall Thanks. Feel free to post an answer so I can accept it. $\endgroup$ – MathematicsStudent1122 Dec 31 '18 at 0:45
  • $\begingroup$ @MathematicsStudent1122 nah you go for it. It’s good for you. $\endgroup$ – Randall Dec 31 '18 at 1:10
  • $\begingroup$ The statement is true however, for every non-trivial 3-group ... $\endgroup$ – Nicky Hekster Dec 31 '18 at 10:34

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