Every Normal Subgroup of An contains Every 3-Cycle in An

Suppose that H is a normal subgroup of An, n > 4 and H contains one three cycle. I prove that H contains every three cycle in An.

let (a,b,c) be in H. Then as n>4 there is a three cycle (a,d,e) so that (a,d,e)(a,b,c)(a,e,d) is in H. (because (a,d,e) is in An and H is normal in An)

then (a,d,e)(a,b,c)(a,e,d) = (a)(b,c)(c,d) = (a)(b,c,d)(e) = (b,c,d)

Now, noting that "d" was arbitrary I can repeat this step and transform (a,b,c) to (x,y,z) for any (x,y,z)

(a,b,c) becomes (b,c,x) becomes (c,x,y) becomes (x,y,z)

Hence any (x,y,z) belongs in H. H contains every three cycle.

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