Wikipedia says that the center of S4 is trivial. Is it?

Question

Wikipedia says that the center of the symmetric group n>=3 is trivial.

https://en.wikipedia.org/wiki/Center_(group_theory)#Examples

But IIRC, S4 is the same group as the rotations of a cube, and the group of rotations of a cube has an abelian subgroup Z2×Z2 consisting of the rotations by 180 degrees, which is nontrivial.

So ... what's going on?

Answer 1

The statement $S_4$ has an abelian subgroup $H=\mathbb{Z}_2\times \mathbb{Z}_2$ means that every element in $H$ commutes with other element in $H$. This does not mean that every element in $H$ commutes with all the elements in $S_4$. So this does not imply $H\leq Z(S_4)$.

Answer 2

Not every abelian subgroup of a group is central. Examples abound.

Let's see, to take a few, any finite group divisible by a prime $p$ has a (cyclic) subgroup of order $p$, by Cauchy's theorem (which is abelian). But that subgroup needn't be central. The $S_n'$s ($n\ge3$) indeed provide infinitely many examples.

So you're confusing the definitions. Central elements commute with every element of the parent group.

And, this may be a little like beating a dead horse now but, by Cayley's theorem, any finite abelian group embeds in an $S_n$.