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I was playing around with the Cayley graphs for some simple groups today and stumbled across something interesting, but can't quite figure out if there's something deeper going on. Here's what I did:

Consider the multiplicative group of the integers mod p, $\mathbb{Z}_{p}^{\times}$. We can generate $\mathbb{Z}_{p}^{\times}$ with any single element and obtain a simple Cayley graph. However, consider the Cayley graph generated by all primes strictly less than p, i.e. let $S=\{[a]: a \text{ is prime and } a and let $\Gamma_{p}$ be the Cayley graph $\Gamma_{p} = (\mathbb{Z}_{p}^{\times}, S)$.

Here is the Mathematica code I was using to generate some of the graphs:

plotGraph[p_] := (   primesN := Table[Prime[n], {n, PrimePi[p] - 1}]; (*get generators*)    (*function to compute adjacency matrix entries*)   f[i_, j_] := (If[MemberQ[Mod[i*primesN, p], j], Return[1]];0);     M := Array[f, {p - 1, p - 1}]; (*create adjacency matrix*)   MatrixForm[M]    GraphPlot3D[M, VertexLabeling -> True]   ) 

I noticed that if n is not prime (and of course, $\mathbb{Z}_{n}-\{[0]\}$ is not a group), then the graph $\Gamma_{n}$ is really not interesting. However, when p is prime, the graphs have some nice structure. p=2 is a point, p=3 is a line segment, p=5 a square, p=7 an octahedron, p=11 looks like a pentagonal antiprism. However, I don't know if there is a pattern, or basically what's going on here. Does anyone have any insight?

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    $\mathbb Z_p^\times$ is always a group (that notation is understood by most people to mean «the set of invertible elements $\mod p$») You probably mean the set of non-zero elements, or something.2012-01-11
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    Kind of beside the point, fixed anyways.2012-01-11
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    Correctness is never beside the point (and you did not fix it :) )2012-01-11
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    Ah! I had assumed $\mathbb{Z}_{n}^{\times}$ was just different notation for the set of nonzero elements. *Now* it's fixed.2012-01-11

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