(a) Let G be a group and XCG. (i) Show that the set F = {(x1x23...n)...

Transcribed Image Text:

(a) Let G be a group and XCG. (i) Show that the set F = {(x1x2®3...n) | n e N, x; E XU X-1, 1 < i < n} with multiplication of G is a subgroup of G. (ii) Prove that F is the smallest subgroup of G containing X. (b) Let n e N. Show that the index of (n) in Z is n. (c) For (1 2), (1 3) E S3, determine whether ((1 2))((1 3)) is a subgroup of S3. (a) Let G be a group and XCG. (i) Show that the set F = {(x1x2®3...n) | n e N, x; E XU X-1, 1 < i < n} with multiplication of G is a subgroup of G. (ii) Prove that F is the smallest subgroup of G containing X. (b) Let n e N. Show that the index of (n) in Z is n. (c) For (1 2), (1 3) E S3, determine whether ((1 2))((1 3)) is a subgroup of S3.