An Is Generated By 3-Cycles

Next I need to show that An is generated by three cycles. I begin with a product of transpositions X of length 2k and induce on k.

Note that every 3-cycle is a product of an even number of transpositions so every three cycle generates a subgroup of A5.

Inducing on the number of transpositions in A5 ( on k in pairs as 2k)

if {a,b} = {c,d} then (a,b)(c,d) = e which is fine eX = X.

Now suppose {a,b}{c,d} have one common member (a,b)(b,c) = (a,b,c)

So (a,b,c)X is generated by 3 cycles by induction as X is so generated by three cycles by the induction hypothesis.

suppose {a,b}{c,d} are disjoint

(a,b)(c,d) = (a,b,c)(b,c,d) So (a,b,c)(b,c,d)X is also generated by three cycles.

The induction is correct, 'An' for n>3 is generated by three cycles.

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