I have a doubt, I do not know if my method of solving this problem is correct. I have to count how many isomorphic subgroups to $ \mathbb{Z}_3\times \mathbb{Z}_3$ there are in $ S_6 $. being abelian I have to find two generators that switch between them, so I think you can choose a three cycle and then I have 40 choices, having the second generator forced (the other three cycle, and its power) then in all 80 choices, dividing by the number of elements of order 3 I have that ultimately there are 20 in the subgroups searched for.
2026-03-30 06:29:20.1774852160
Number of subgroups of $S_6$ isomorphic to $C_3\times C_3$
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