Let G be a group of odd order. a) Prove that if x E G, then (x2) = (x). b) Showthatthemapp:G-Gsuchthatp(g)=g2 isabijection. c) If x and y are elements of G such that yxy-1 = x-1, show that x and y2 commute and that x = =1.
Let G be a group of odd order. a) Prove that if x E G, then (x2) = (x). b) Showthatthemapp:G-Gsuchthatp(g)=g2 isabijection. c) If x and y are elements of G such that yxy-1 = x-1, show that x and y2 commute and that x = =1.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.6: Quotient Groups
Problem 30E: Let G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.
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