Mathematics

An Is Generated By 3Cycles
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 3cycle 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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