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A group G is a simple group if the only normal subgroups of G are G...

A group G is a simple group if the only normal subgroups of G are G itself and {e}. In other words, G is simple if G has no non-trivial proper normal subgroups.

Algebraists have proven (using more advanced techniques than ones we’ve discussed) that An is a simple group for n ≥ 5.

Using this fact, prove that for n ≥ 5, An has no subgroup of order n!/4 .

(This generalizes HW5,#3 as well as our counterexample from class as to the converse of Lagrange’s Theorem.)

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