Abstract Nonsense

Crushing one theorem at a time

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


Point of post: This is a continuation of this post.

From this we can derive some interesting results, but first we need some notation. For any group G (not necessarily a p-group) and any H\subseteq G we call the set

\text{ }

\displaystyle \left\{g\in G:gHg^{-1}=H\right\}

\text{ }

(note that H need only be a subset) the normalizer of H and denote this N\left(H\right). It’s clear that N(H) is a subgroup of H and if H itself is group it’s the largest (by set containment) normal group containing H. With this in mind we have:

Theorem: Let G be a group such that |G|=p^n where p is prime and n\in\mathbb{N}. Then:

\text{ }

\displaystyle \begin{aligned}&\mathbf{(1)}\quad\mathcal{Z}\left(G\right)\textit{ is non-trivial}\\ &\mathbf{(2)}\quad\mathcal{Z}\left(G\right)\cap N\textit{ is non-trivial for any non-trivial }N\unlhd G\\ &\mathbf{(3)}\quad \textit{If }H<G\textit{ then }H\subsetneq N(H)\textit{ and thus if}\left|H\right|=p^{n-1}\textit{ then }H\unlhd G\end{aligned}

\text{ }

Proof:

\mathbf{(1)}: Let \Gamma be a set of one representative from each conjugacy class of G with more than one element. By the class equation we have that

\text{ }

\displaystyle p^n-\sum_{r\in\Gamma}\left(G:G_r\right)=\left|\mathcal{Z}\left(G\right)\right|

 \text{ }

note though that for each r\in\Gamma we have that \left(G:G_r\right)\left|G_r\right|=p^n so that \left(G:G_r\right)\mid p^n. Thus,

\text{ }

\displaystyle p\mid \sum_{r\in\Gamma}\left(G:G_r\right)

\text{ }

and thus consequently

\text{ }

\displaystyle p\mid p^n-\sum_{r\in\Gamma}\left(G:G_r\right)=\left|\mathcal{Z}\left(G\right)\right|

 \text{ }

and since \left|\mathcal{Z}\left(G\right)\right|\geqslant 1 it follows that \left|\mathcal{Z}\left(G\right)\right|=pn for some n\in\mathbb{N}. The conclusion follows.

\text{ }

\mathbf{(2)}: Recall from our proof the class equation that

\text{ }

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

 \text{ }

and thus

\text{ }

\displaystyle N=\left(\mathcal{Z}\left(G\right)\cap N\right)\;\uplus\;\biguplus_{r\in\Gamma}\left(N\cap C(r)\right)

\text{ }

and thus

\text{ }

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

\text{ }

notice though that if x\in N then

\text{ }

\displaystyle C(x)=\left\{gxg^{-1}:g\in G\right\}\subseteq gNg^{-1}=N

\text{ }

(since N\unlhd G) and if x\notin N then C(x)\cap N=\varnothing. It follows then that

\text{ }

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

\text{ }

but using the same logic as before, noting that |N|\mid p^n, we may conclude that p\mid\left|\mathcal{Z}\left(G\right)\right| and thus \mathcal{Z}\left(G\right) is non-trivial.

\text{ }

\mathbf{(3)}: This last result follows by noticing that \left(G:H\right)=p which is the smallest prime dividing |G| and the conclusion follows from an earlier result.

\blacksquare

\text{ }

Corollary: Every group of order p^2 where p is a prime is abelian.

Proof: By the above theorem we have that \mathcal{Z}\left(G\right) is non-trivial and so by Lagrange’s Theorem we have that \displaystyle \left|\mathcal{Z}\left(G\right)\right|=p,p^2. Assume that \left|\mathcal{Z}\left(G\right)\right|=p then \left|G/\mathcal{Z}\left(G\right)\right|=p and thus (from previous theorem) we may conclude that G/\mathcal{Z}\left(G\right) is cyclic. But, from an earlier theorem we may then conclude that G is abelian and thus \left|\mathcal{Z}\left(G\right)\right|=p^2 which is a contradiction. Thus, it follows that \left|\mathcal{Z}\left(G\right)\right|=p^2 and thus \mathcal{Z}\left(G\right)=G from where the conclusion follows. \blacksquare

\text{ }

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


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January 6, 2011 - Posted by | Algebra, Group Theory | , , , ,

5 Comments »

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