Find all abelian subgroups of $A_4 \times S_3$.

Question

I don't really know how to start this question other than listing the group out, however I feel that would be missing the point of the question somewhat.

Any direction would be greatly appreciated!

Answer 1

The highest possible order of an element $(a,b)\in A_4\times S_3$ would be $\operatorname{lcm} (\vert a\vert,\vert b\vert) \le6$.

By FTFAG, the only possibilities, up to isomorphism, are: $\Bbb Z_2, \Bbb Z_3, \Bbb Z_3\times \Bbb Z_3, \Bbb Z_2\times \Bbb Z_2,\Bbb Z_6, \Bbb Z_6\times\Bbb Z_6, \Bbb Z_3\times \Bbb Z_6, \Bbb Z_2\times \Bbb Z_6. $.

It remains to check which of these are subgroups. For instance, $\langle ((123),(12))\rangle \cong\Bbb Z_6$. The are "other" $\Bbb Z_6$'s.

There are obviously $\Bbb Z_2$'s and $\Bbb Z_3$'s.

$\langle ((123),e),(e, (123)) \rangle \cong\Bbb Z_3\times \Bbb Z_3$.

There is a Klein four group since $A_4$ has one.

Etc. There's some work left.