1. The problem statement, all variables and given/known data A. Let |g| = 20 in a group G. Compute |g^2|, |g^8|,|g^5|, |g^3| B. In each case find the subgroup H = <x,y> of G. a) G = <a> is cyclic, x = a^m, y = a^k, gcd(m,k)=d b) G=S_3, x=(1 2), y=(2 3) c) G = <a> * <b>, |a| = 4, |b| = 6, x = (a^2, b), y = (a,b^3) 3. The attempt at a solution A. I know |g^2| = 20/2 = 10 and |g^5| = 20/5 = 4 But |g^8|, |g^3| don't know.. B. a)H=<a^d> , right? but I don't know how to solve b) and c) Thanks!
Don't forget that if [itex]g^{20}=e[/itex] then [itex]g^{40}=e[/itex] also. Yes. b should be easy, because you've got a concrete group to play with. Just get in there and start computing. As for c, what does <a>*<b> mean?
The least common multiple of 20 and 8 is 2*4*5= 40. [itex](g^8)^5= (g^20)^2= e[/itex]. The least common multiple of 3 and 20 is 60. [itex](g^3)^20= (g^20)^3= e[/itex].