Thread: Groups

two questions. First, what is a group of transformations? I know what it is, but how do i enumerate the elements? i.e, let T be the set of all functions such that f(x) = x, 2x, what? The question asks me to find a group of transformations isomorphic to the group of integers mod 8 under addition. I know what that means. it means find a function,phi, that is a bijection between my group and the transformation group. BUT what is that function if i don't even know the elements in the transformation group. can i use the function in the trans. group for my phi or what?


second question: if a cyclic group G is generated by "a" of order m, prove that the powers of a^k generate all of G iff gcd (k,m)=1. I just need help with it. or a hint.

o(G)=m so a^m=e the identy of G. As (k,m)=1 therefore k and m are relativly prime so order of a^k is m also i.e (a^k)^m=e which implies the cyclic group generated by a^k is the whole G and not its subgroup. In particular suppose m=12 and k= 5 then a^12=e and a^60=(a^5)^12=e as (5,12)=1, in this case the only cyclic subgroups of G will be of order 2, 3, 4,6 generated possibly by a^6, a^4, a^3 and a^2 respectively.

Originally Posted by Zero266

two questions. First, what is a group of transformations? I know what it is, but how do i enumerate the elements? i.e, let T be the set of all functions such that f(x) = x, 2x, what? The question asks me to find a group of transformations isomorphic to the group of integers mod 8 under addition. I know what that means. it means find a function,phi, that is a bijection between my group and the transformation group. BUT what is that function if i don't even know the elements in the transformation group. can i use the function in the trans. group for my phi or what?

Is the group of transformations just any old transformations? Or is it transformations between specific sets? Please provide more details. What exactly is the group we are working with? If you're taking about all transformation between all sets, then clearly all such transformations are not countable (i.e. cannot be put in bijection with the natural numbers).