Paper 3, Section II, 7D7 \mathrm{D}

Groups | Part IA, 2021

Let GG be a finite group of order nn. Show that GG is isomorphic to a subgroup HH of SnS_{n}, the symmetric group of degree nn. Furthermore show that this isomorphism can be chosen so that any nontrivial element of HH has no fixed points.

Suppose nn is even. Prove that GG contains an element of order 2 .

What does it mean for an element of SmS_{m} to be odd? Suppose HH is a subgroup of SmS_{m} for some mm, and HH contains an odd element. Prove that precisely half of the elements of HH are odd.

Now suppose n=4k+2n=4 k+2 for some positive integer kk. Prove that GG is not simple. [Hint: Consider the sign of an element of order 2.]

Can a nonabelian group of even order be simple?

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