3.II .5E. 5 \mathrm{E} \quad

Groups | Part IA, 2008

For a normal subgroup HH of a group GG, explain carefully how to make the set of (left) cosets of HH into a group.

For a subgroup HH of a group GG, show that the following are equivalent:

(i) HH is a normal subgroup of GG;

(ii) there exist a group KK and a homomorphism θ:GK\theta: G \rightarrow K such that HH is the kernel of θ\theta.

Let GG be a finite group that has a proper subgroup HH of index nn (in other words, H=G/n)|H|=|G| / n). Show that if G>n|G|>n ! then GG cannot be simple. [Hint: Let GG act on the set of left cosets of HH by left multiplication.]

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