Question

icu lact that thie muiiplication of matrices with real coefficients is associative. 1. Show that every cyclic group is abelian. 2. Let m be an integer greater than 1. For any l e Z, let Im denote the remain- der of the division of I by m. Let: Zm = {0. 1, 2, . . . ,m-1)- on Zm as follows: Define a binary operation where + is the usual addition of integers. (a) Show that G (Zm,*) is a group. (b) Show that for any j E G, we have: TT7L ord j ged(m,J) (c) Show that G = (225, *) is cyclic, and find all of its generators. 3·Let SL(2, Z) denote the set of 2 x 2 matrices with integer coefficients whose matrix multiplica- determinants are equal to 1. Is SL(2,Z) a group under

I am not sure how to do

Answer 1

Answer-1. Let $G$ be a cyclic group. Then, there exists $a\in G$ such that $G=<a>=\{a^m:m\in\mathbb{Z}\}.$ Now, let $x,y\in G.$ Then, there exist $m,n\in \mathbb{Z}$ such that $x=a^m$ and $y=a^n.$ Now,$xy=a^ma^n=a^{m&plus;n}=a^{n&plus;m}=a^na^m=yx$ and hence it follows that for any $x,y\in G,$$xy=yx$, that is, the cyclic group $G$ is abelian.