An (n>4) Is A Simple Group

Theorem: An is simple for n>4

Suppose H is a normal subgroup of An, and H is not {e} Then H contains a three cycle.

Therefore H contains every three cycle of An

Then, every element of An is a product of three cycles, An being generated by three cycles.

So, An is a subgroup of or is equal to H (H contains An)

therefore since H is contained in An, H = An (H is isomorphic to An)

An is simple for n>4.

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