Consider group G and its subgroup H (H ≤ G): [G : H] = 2. Prove why H is a normal subgroup of G.
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Consider group G and its subgroup H (H ≤ G): [G : H] = 2. Prove why H is a normal subgroup of G.
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- 18. If is a subgroup of , and is a normal subgroup of , prove that .With H and K as in Exercise 18, prove that K is a normal subgroup of HK. Exercise18: If H is a subgroup of G, and K is a normal subgroup of G, prove that HK=KH.Find the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.
- 19. With and as in Exercise 18, prove that is a subgroup of . Exercise18: 18. If is a subgroup of , and is a normal subgroup of , prove that .16. Let be a subgroup of and assume that every left coset of in is equal to a right coset of in . Prove that is a normal subgroup of .27. Suppose is a normal subgroup of order of a group . Prove that is contained in , the center of .
- 18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.Suppose G1 and G2 are groups with normal subgroups H1 and H2, respectively, and with G1/H1 isomorphic to G2/H2. Determine the possible orders of H1 and H2 under the following conditions. a. G1=24 and G2=18 b. G1=32 and G2=40Let be a subgroup of a group with . Prove that if and only if .
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