Abstract algebra-prove some results-subgroups

Reference no: EM13117496

1. Define (C_G)(H) = {g is a number in G: g h = h g for all h is a number in H), where H is a subgroup of the group G. Prove that (C_G)(H) is a subgroup of G. Note: (C_G)(H) is called the centralizer of H in G.

2. Define (N_G)(H) = {g is a number in G: gH = Hg], where H is a subgroup of the Group G. Prove that (N_G)(H) is a subgroup of G. Note: (N_G)(H) is called the normalizer of H in G.

3. Prove that (C_G)(H) is a normal subgroup of (N_G)(H), where H is a subgroup fo the group G.

4. Let G be a subgroup such that |G| = p q, where p and q are primes and p < q. Prove that G must have a normal subgroup.

Reference no: EM13117496

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