For n at least 5, the commutator subgroup of Sym(n) is Alt(n) (via Project Crazy Project)

Prove that the commutator subgroup of S_n is A_n for all n geq 5. We know that [S_n,S_n] is characteristic, hence normal, in S_n. Since n geq 5, then, this commutator is either 1, A_n, or S_n. Now [S_n,S_n] neq 1 since S_n is nonabelian. Moreover, since S_n/A_n cong Z_2 is abelian, we have that [S_n, S_n] leq A_n. Thus [S_n,S_n] = A_n. … Read More

via Project Crazy Project

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i love the world of Peace
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