Mapping Homomorphisms to Commutative n-Tuples: A Bijection

[daswerth]

Homework Statement

Consider the set $$Hom$$ of homomorphisms from $$\mathbb{Z}^n$$ (the n-dimensional integer lattice) to a group $$G$$.

Also let $$S = \left\{ \, ( g_1, g_2, \dots, g_n ) \, | \, g_i g_k = g_k g_i, \text{where} \, 0 < i,k \leq n, g_i \in G \right\}$$, the set of n-tuples from G which consist only of elements that commute with each other.

Task: Produce a natural bijection from $$Hom$$ to $$S$$.

Homework Equations

The Attempt at a Solution

An example of a homomorphism from $$\mathbb{Z}^n$$ to $$G$$ would be to take $$\phi (X) = \phi ( x_1 e_1 + x_2 e_2 + \dots + x_n e_n ) = \phi (e_1)^{x_1} \phi (e_2)^{x_2} \dots \phi (e_n)^{x_n}$$. For each $$e_i$$ (unit vector) we associate an element of $$G$$, so $$\phi (e_i) = g_i$$. In order for this to be a homomorphism, we need to have $$\phi (X) \phi (Y) = \phi (X+Y)$$. This means each of the $$g_i$$ must commute with each other. In other words, we associate the unit vectors, $$e_1 \dots e_n$$, with the elements of an n-tuple from $$S$$.

I just don't see how to uniquely assign a homomorphism from $$Hom$$ to an n-tuple from $$S$$. That's what I'm stuck on. Once that light clicks on, I'm confident I can show that it's bijective. So, your hints will be very much appreciated!

This problem is from an undergraduate Algebra class. Thanks!

 

[daswerth]

It just occurred to me that this may be simpler than I thought. The obvious mapping would be simply to send the homomorphism to the n-tuple that it assigns its unit vectors to. I had assumed the map would be more complicated/interesting.

I'm going to investigate whether this map is bijective.