Normal Subgroups of the Symmetric Group (4.6.2)

Dummit and Foote Abstract Algebra, section 4.6, exercise 2:

Find all normal subgroups of S_n for n > 4.

Proof: Assume a proper nontrivial normal subgroup N such that N ≠ A_n (which, due to A_n's maximality, implies A_n is not contained in N). We have N∩A_n is normal in A_n since, for all x∈N∩A_n and a∈A_n, axa^-1∈N (due to N's normality) and axa^-1∈A_n. Therefore, due to A_n's simplicity, we have N∩A_n = 1, so that every nonidentity element of N is of order 2 (since σ²∈A_n) and not in A_n. Assume | N | > 2 such that there are distinct nonidentity x and y (that are their own inverses). Since xy∈A_n, this means xy = 1, so x = y, a contradiction. Therefore |N| = 2, and since N is a union of conjugacy classes and every nonidentity element has some distinct nonidentity conjugate since C_Sn(σ) < S_n due to Z(S_n) = 1, this exhausts all possibilities for N. Thus, the only proper nontrivial normal subgroup in S_n for n > 4 is A_n. Of course, 1 and S_n are also normal in S_n.

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