Abelian group Definition and 57 Threads

Homework Statement G=(Z+Z+Z)/N where Z denote the integers and + is direct sum and N = <(7,8,9), (4,5,6), (1,2,3)> or the smallest submodule of Z+Z+Z containing these 3 vectors. How would you describe G? The Attempt at a Solution N = {a(7,8,9)+b(4,5,6)+c(1,2,3)|a,b,c in Z} = {(7a+4b+c...

Please HELP! So, I have to go about proving the following, but I have no idea where to even start: I. Let S = R – {3}. Define a*b = a + b – (ab)/3. 1. Show < S,*> is a binary operation [show closure]. 2. Show < S,*> is a group. 3. Find *-inverse of 11/5 II. Let G be a group with x,y...

I'm working on a proof for subgroups of free abelian groups and am having trouble with a step (I know other methods, but would like to try and make this one work if possible). The basic idea is let G be a free abelian group with generators (g_1...g_n) and let H be a subgroup of G. Assuming a...

Hello. I was wondering how I could prove if a set of numbers along with some arbitrary operation is an abelian group.

Basically, I have to show an example such that for a nonabelian group G, with a,b elements of G, (a has order n, and b has order m), it is not necessarily the case that (ab)^mn= e. where e is the identity element. im not sure where to start. =\

let G be an abelian group, and n positive integer phi is a map frm G to G sending x->x^n phi is a homomorphism show that a.)ker phi={g from G, |g| divides n} b.) phi is an isomorphism if n is relatively primes to |G| i have no clue how to even start the prob...:-(

I've got a question. It pertains to a proof I'm doing. I ran into this stumbling block. If I could show this I think I could complete the proof. G is a finite Abelian Group such that there exits more than one element of order 2 within the group. more than one element of the form b not...