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I have to find all the conjugancy classes of $A_4$ when $y=(234)$ is a cycle and the transposition $x=(12)(34)$ in $S_4$ which satisfies $y^3=x^2=(xy)^3=I$, is there any clever tricks/shortcurts or do I have to "brute force" to find all the conjugacy classes by taking $\{x\}=\{gxg^{-1}\mid \forall g\in G\}$

S.G.
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user8700908
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  • For your particular $x$ and $y$ it is already true that $y^3=x^2=(xy)^3=I.$ So why even say that? You certainly don't mean these equations true for all choices of $x,y.$ – coffeemath Mar 04 '21 at 13:00
  • Possible duplicate: https://math.stackexchange.com/questions/1899500/problem-involving-conjugacy-classes-of-the-alternating-group – user326159 Mar 04 '21 at 13:37
  • Please make clear what is given, which is the question, and which are your own attempts. This would be best realized by building at least three sentences, one for each issue. (Regarding the own attempts, use as many sentences as needed.) – dan_fulea Mar 04 '21 at 13:50
  • (The cycle type of a permutation is a good way to spot the elements of a conjugacy class, the possible cycle types give the conjugacy classes.) – dan_fulea Mar 04 '21 at 13:51

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