Prove that every non-trivial normal subgroup H of A5​ contains a 3 -cycle. (Hint: The 3 -cycles are the non-identity ele

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Prove That Every Non Trivial Normal Subgroup H Of A5 Contains A 3 Cycle Hint The 3 Cycles Are The Non Identity Ele 1 (49.1 KiB) Viewed 36 times

Prove that every non-trivial normal subgroup H of A5​ contains a 3 -cycle. (Hint: The 3 -cycles are the non-identity elements of A5​ with the largest number of fixed points. If σ∈Sn​, a reasonable way of trying to construct a permutation out of σ with more fixed points than σ is to form a commutator [σ,τ]=στσ−1τ−1 for an appropriate permutation τ∈Sn​. This idea is used in the solution of Rubik's cube. Why is this a reasonable thing to try?)