Paper 3, Section II, D

Groups | Part IA, 2011

(a) Let GG be a finite group and let HH be a subgroup of GG. Show that if G=2H|G|=2|H| then HH is normal in GG.

Show that the dihedral group D2nD_{2 n} of order 2n2 n has a normal subgroup different from both D2nD_{2 n} and {e}\{e\}.

For each integer k3k \geqslant 3, give an example of a finite group GG, and a subgroup HH, such that G=kH|G|=k|H| and HH is not normal in GG.

(b) Show that A5A_{5} is a simple group.

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