Alt(n) is generated by the set of all 3-cycles

Prove that A_n is generated by the set of all 3-cycles for n \geq 3.


Let n \geq 3 and let T = \{ (a\ b\ c) \ |\ 1 \leq a,b,c \leq n \} be the set of 3-cycles in A_n.

Note that A_n contains T, so that \langle T \rangle \leq A_n.

Recall that A_n consists of all permutations which can be written as an even product of transpositions; more specifically, A_n is generated by the set of all products of two distinct 2-cycles. (Here we use the fact that n \geq 3.) Each product \sigma of two 2-cycles has one of two forms.

If \sigma = (a\ b)(c\ d), note that \sigma = (a\ c\ b)(a\ c\ d).

If \sigma = (a\ b)(a\ c), note that \sigma = (a\ c\ b).

Thus A_n \leq \langle T \rangle, hence A_n = \langle T \rangle.

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