Abstract Algebra: Prove Some Results About Subgroups
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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.
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Solution Summary
Four problems in group theory are solved. The first one shows that the centralizer C(H) of a subgroup H in G is a subgroup of G, the second one shows that the normalizer N(H) is a subgroup in G, the third one shows that C(H) is a normal subgroup of N(H), and the fourth one shows that if the order of G is a product to two primes, then G has a normal subgroup. Solution is in a .pdf attachment.
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