Ιs $A_4 \times \mathbb{Z}_3$ isomorphic to $S_3 \times S_3$?
I am trying to find an element of $S_3 \times S_3$ which has an order, let's say $a$, and $A_4 \times \mathbb{Z}_3$ has no element of such order.
abstract-algebrafinite-groupsgroup-theorysymmetric-groups
Ιs $A_4 \times \mathbb{Z}_3$ isomorphic to $S_3 \times S_3$?
I am trying to find an element of $S_3 \times S_3$ which has an order, let's say $a$, and $A_4 \times \mathbb{Z}_3$ has no element of such order.
Best Answer
In $S_3\times S_3$ there are
while in $A_4\times\mathbb{Z}_3$ there are
so the easiest way to state $S_3\times S_3 \not\simeq A_4\times\mathbb{Z}_3$ is probably to compare the number of involutions (elements with order $2$) in both groups. Or wonder about the largest abelian subgroups, which in the former case are isomorphic to $\mathbb{Z}_3\times\mathbb{Z}_3$ and in the latter case are isomorphic to $\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_3$.