isomorphism of relatively prime groups

PennState666

1. The problem statement, all variables and given/known data
Allow m,n to be two relatively prime integers. You must prove that Z(sub mn) ≈ Z(sub m) x Z(sub n)


2. Relevant equations
if two groups form an isomorphism they must be onto, 1-1, and preserve the operation.


3. The attempt at a solution
since m and n are relatively prime, the gcd(m, n) = 1.
mhm, very stumped from the start.

jbunniii

What do you know about the group Z(sub mn)? What are some facts about it? Is it abelian, etc.?

PennState666

Isomorphisms isnt my thing, I do not know many facts about Z(sub mn). I can say the contents of Z(sub mn) are {[0],[1], [2],.... [mn]}. generated by a single element? Im not sure...Abelian? yes i believe so...

jbunniii

Yes, generated by a single element is the key. In other words, it's a cyclic group. (Side note: cyclic groups are always abelian, but you don't need that fact for this problem.)

So if Z(sub mn) is isomorphic to Z(sub m) x Z(sub n), then that group would also have to be cyclic, right? Can you show that Z(sub m) x Z(sub n) is cyclic?

PennState666

mhmm i believe so, but i dont know where to start since I am dealing with a cross product...

Office_Shredder

If you had to guess at which element of Z_mxZ_n generated the whole group, what would it be?

If you're not sure, try some small examples like Z₂xZ₃ or Z₃xZ₄ and see what you can come up with

PennState666

What would the elements of Z2 x Z3 look like again?

kru_

Ordered pairs (a, b) such that a is from Z2, and b is from Z3.

jbunniii

Well, if A and B are any sets (whether or not they have a group structure), then what is the definition of A x B? This is called the Cartesian product of A and B. The elements are simply ordered pairs of the form [itex](a,b)[/itex], where [itex]a \in A[/itex] and [itex]b \in B[/itex].

And if A and B are groups, then there's a natural way to define a group operation on A x B.

Surely this is discussed in your textbook or lecture? There's no way you can proceed with this problem without first understanding what the group is.

PennState666

yes of course I just wanted to be refreshed and reassured...
Z2 x Z3 = {([0],[0]), ([0],[1]), ([0],[2]), ([1],[0]), ([1],[1]), ([1],[2])}...
I cannot grasp how this could be generated by a single element though?

jbunniii

Look at the six elements. In order for the group to be generated by one element, clearly that element can't have a [0] in either of the two positions of the ordered pair. So that only leaves two candidates: either ([1],[1]) or ([1],[2]). Why not try each and see if it works? Just add it to itself over and over and see if you can generate the group.

PennState666

the generator for the group Z(sub 2) x Z(sub 3) = <[1],[1]> and since the operation is addition, the generator for a group in the form Zn x Zm = <[1], [1]>. so Zn x Zm is cyclic and abelian.

jbunniii

OK, good. Note that <[1],[2]> also generates the group.

So Z2 x Z3 is cyclic, and I assume you observed that it has order 6. That looks promising, because Z6 is also cyclic with order 6.

Now you need to define an isomorphism [itex]\phi: Z_{6} \rightarrow Z_2 \times Z_3[/itex] between the groups. As a first step, I suggest choosing a generator [itex]g[/itex] for [itex]Z_6[/itex] and defining [itex]\phi(g)[/itex].