let x be a nonempty G-set and let You be a nonempty subset of x. let Gy={g € G | gy=y for all your €}.prove that Gy is a subgroup of G
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let x be a nonempty G-set and let You be a nonempty subset of x. let Gy={g € G | gy=y for all your €}.prove that Gy is a subgroup of G
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- 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 .If H and K are arbitrary subgroups of G, prove that HK=KH if and only if HK is a subgroup of G.22. If and are both normal subgroups of , prove that is a normal subgroup of .
- 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.44. Let be a subgroup of a group .For, define the relation by if and only if . Prove that is an equivalence relation on . Let . Find , the equivalence class containing .18. If is a subgroup of , and is a normal subgroup of , prove that .
- Let be a group of order 24. If is a subgroup of , what are all the possible orders of ?23. Prove that if and are normal subgroups of such that , then for all(See Exercise 26) Let A be an infinite set, and let H be the set of all fS(A) such that f(x)=x for all but a finite number of elements x of A. Prove that H is a subgroup of S(A).
- Let be a subgroup of a group with . Prove that if and only if .Let A be a given nonempty set. As noted in Example 2 of section 3.1, S(A) is a group with respect to mapping composition. For a fixed element a In A, let Ha denote the set of all fS(A) such that f(a)=a.Prove that Ha is a subgroup of S(A). From Example 2 of section 3.1: Set A is a one to one mapping from A onto A and S(A) denotes the set of all permutations on A. S(A) is closed with respect to binary operation of mapping composition. The identity mapping I(A) in S(A), fIA=f=IAf for all fS(A), and also that each fS(A) has an inverse in S(A). Thus we conclude that S(A) is a group with respect to composition of mapping.27. Suppose is a normal subgroup of order of a group . Prove that is contained in , the center of .