14 lines
519 B
TeX
14 lines
519 B
TeX
\section{Applications of Factor Groups}
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\begin{theorem}[$\mathbf{G/Z}$ Theorem]
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Let $G$ be a group and let $Z(G)$ be the center of $G$. If $G/Z(G)$ is cyclic, then $G$ is Abelian.
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\end{theorem}
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\begin{theorem}[$\mathbf{G/Z(G) \approx \text{Inn}(G)}$]
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For any group $G$, $G/Z(G)$ is isomorphic to Inn$(G)$.
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\end{theorem}
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\begin{theorem}[Cauchy's Theorem for Abelian Groups]
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Let $G$ be a finite Abelian group and let $p$ be a prime that divides the order of $G$. Then $G$ has an element of order $p$.
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\end{theorem}
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