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Homomorphisms of a group

  • Thread starter daswerth
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  • #1
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Homework Statement



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

Also let [tex] 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\}[/tex], the set of n-tuples from G which consist only of elements that commute with each other.

Task: Produce a natural bijection from [tex] Hom [/tex] to [tex] S [/tex].

Homework Equations





The Attempt at a Solution



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

I just don't see how to uniquely assign a homomorphism from [tex] Hom [/tex] to an n-tuple from [tex] S [/tex]. 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!
 

Answers and Replies

  • #2
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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.
 

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