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?
Best Answer
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 $H\leq Z(S_4)$.