2. Suppose an element a E G has order 2. Prove that (a) is a normal subgroup of G if and only if a is in the center of G.
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- Find the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.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.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.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 .
- Let H be a torsion subgroup of an abelian group G. That is, H is the set of all elements of finite order in G. Prove that H is normal in G.Show that An has index 2 in Sn, and thereby conclude that An is always a normal subgroup of Sn.Prove or disprove that H={ [ 1a01 ]|a } is a normal subgroup of the special linear group SL(2,).