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

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.

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