# Abstract Nonsense

## Review of Group Theory: Group Actions (Pt. IV Conjugation and the Class Equation)

Point of post: In this post we apply our methods of group actions to the particular group action of conjugation. We use this to derive the class equation and from this derive several key results the a corollary of which will be that every group of order $p^2$ with $p$ prime is abelian.

Motivation

In our last few posts we’ve spent a considerable amount of time developing the theory of group actions, orbits, orbit decompositions, etc. Now we’re going to put it to some real use. We’re going to consider a group $G$ acting on itself via conjugation. In other words, we’ll say that $g\cdot g'=gg'g^{-1}$. From this we’ll be able to use the Orbit Decomposition Theorem to derive the class equation which states that if $G$ is a finite group then

$\displaystyle |G|=\left|\mathcal{Z}\left(G\right)\right|+\sum_{r\in\Gamma}\left(G:G_r\right)$

where $G_r$ are the isotropy subgroups under the action of conjugation and $\Gamma$ is a set of representatives from each conjugacy class (i.e. the orbits under the action of conjugation) with more than one element. From this we’ll derive several important results including that if $G$ is a finite group with $|G|=p^n$ where $p$ is prime then $\mathcal{Z}(G)$ is non-trivial. As a corollary of this we will be able to conclude that every group of order $p^2$ where $p$ is prime is abelian.

Conjugation

Let $G$ be a group and define the function

$\displaystyle \cdot:G\times G\to G:(g,h)\mapsto ghg^{-1}$

then $\cdot$ is a $G$-action on $G$. Indeed

$\displaystyle g_1\cdot(g_2\cdot h)=g_1\cdot g_2hg_2^{-1}=g_1g_2h(g_1g_2)^{-1}=(g_1g_2)\cdot h$

and

$e\cdot h=ehe^{-1}=h$

We call this action $G$ acting on itself by conjugation. For this particular action there are names for the orbits and isotropy subgroups. In particular, the isotropy subgroup of some $r\in G$ is called the normalizer of $r$ and is denoted $N(r)$ (or $N_G(r)$ when the group under which we’re considering the action may be ambiguous) . The orbit of $r$ under the conjugation action is called the centralizer of $r$ and is denoted $C(r)$ (or $C_G(r)$ for the same reasons) and is called the conjugacy class of $r$. We note the trivial fact that $g\in\mathcal{Z}(G)$ (where $\mathcal{Z}(G)$ is the center of $G$) if and only if $C(g)=\{g\}$.

With this in mind we may now derive the class equation

Theorem: Let $G$ be a finite group then

$\displaystyle \left|G\right|=\left|\mathcal{Z}\left(G\right)\right|+\sum_{r\in\Gamma}\left(G:N(r)\right)$

where $\Gamma$ is a set of representatives from the conjugacy classes which contain more than one element. If $\Gamma=\varnothing$ this sum is taken to be zero.

Proof: By the Orbit Decomposition Theorem we know that

$\displaystyle G=\biguplus_{r\in\Lambda}C(r)$

where $\Lambda$ is a set of representatives from each conjugacy class of $G$. But, we can rewrite this as

$\displaystyle G=\biguplus_{g\text{ such that }\#\left(C(g)\right)=1}C(g)\;\uplus\;\biguplus_{\substack{r\in\Gamma\\ |C(r)|>1}}C(r)$

By previous comment though

$\displaystyle \biguplus_{g\text{ such that }\#\left(C(g)\right)=1}C(g)=\mathcal{Z}(G)$

and thus if $\Gamma$ is the set of all $r\in\Lambda$ such that $\#(C(r))>1$ then

$\displaystyle G=\mathcal{Z}(g)\;\uplus\;\biguplus_{r\in\Gamma}C(r)$

Thus,

$\displaystyle \left|G\right|=\left|\mathcal{Z}(G)\right|+\sum_{r\in\Gamma}\#\left(C(r)\right)$

but, using the fact that $\#\left(C(r)\right)=\left(G:N(r)\right)$ we may conclude that

$\displaystyle \left|G\right|=\left|\mathcal{Z}\left(G\right)\right|+\sum_{r\in\Gamma}\left(G:N(r)\right)$

and the conclusion follows. $\blacksquare$

References:

1. Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. Basic Abstract Algebra. Cambridge: Cambridge UP, 1994. Print.

2. Lang, Serge. Undergraduate Algebra. 3rd. ed. Springer, 2010. Print.

3. Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 200

January 6, 2011 -

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