e-cata Posted April 8, 2012 Posted April 8, 2012 Quote Let G be any group, and C'=\{a \in G: (ax)^{2}=(xa)^{2} for every x \in G\}.Prove that C' is a subgroup of G. This exercise is from chapter 5 of A Book Of Abstract Algebra. I've been studying this book by reading, so this is not homework. Please help. Here is my attempt: If C' is a subgroup, (abx)^{2}=(xab)^{2} and (a^{-1}x)^{2}=(xa^{-1})^{2} for every a and b in C'. It is obvious that Z(G) is included in C'. If we have a group K=\{a,b \in K: a=a^{-1}, ab \neq ba \}, then Z(K)=\emptyset even though (ab)^{2}=(ba)^{2} for every element in K. Thus C' is not necessarily Abelian. 1
Phi for All Posted April 8, 2012 Posted April 8, 2012 ! Moderator Note One thread per question, please. Thanks!
e-cata Posted April 8, 2012 Author Posted April 8, 2012 Solved (a(bx))^{2}=((bx)a)^{2}=(b(xa))^{2}=((xa)b)^{2}=(xab)^{2} and (a^{-1}x)^{2}=(a^{-1}xa^{-1}a)=((a^{-1}xa^{-1})a)^{2}=(a(a^{-1}xa^{-1}))^{2}=(xa^{-1})^{2}. Thus C' is closed with respect to multiplication and inverses, and is a subgroup of G. 1
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