# Homomorphisms of a group

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 $$S$$.

## 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.