16 lines
552 B
TeX
16 lines
552 B
TeX
\section{The Class Equation}
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\begin{corollary}[Class Equation]
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For any finite group $G$,
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\[ \abs{G} = \sum \abs{G:C(a)} \]
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where the sum runs over one element of $a$ from each conjugacy class of $G$.
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\end{corollary}
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\begin{theorem}[$\mathbf{p}$-Groups Have Nontrivial Centers]
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Let $G$ be a nontrivial finite group whose order is a power of a prime $p$. Then $\Z(G)$ has more than one element.
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\end{theorem}
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\begin{corollary}[Groups of Order $\mathbf{p^2}$ Are Abelian]
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If $\abs{G}=p^2$, where $p$ is prime, then $G$ is Abelian.
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\end{corollary}
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