14 lines
728 B
TeX
14 lines
728 B
TeX
\section{The First Isomorphism Theorem}
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\begin{theorem}[First Isomorphism Theorem (Jordan, 1870)]
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Let $\phi$ be a group homomorphism from $G$ to $\overline{G}$. Then the mapping from $G/\ker\phi$ to $\phi(G)$, given by $g\ker\phi \to \phi(g)$, is an isomorphism. In symbols, $G/\ker\phi \approx \phi(G)$.
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\end{theorem}
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\begin{corollary}
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If $\phi$ is a homomorphism from a finite group $G$ to $\overline{G}$, then $\abs{\phi(G)}$ divides $\abs{G}$ and $\abs{\overline{G}}$.
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\end{corollary}
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\begin{theorem}[Normal Subgroups Are Kernels]
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Every normal subgroup of a group $G$ is the kernel of a homomorphism of $G$. In particular, a normal subgroup $N$ is the kernel of the mapping $g \to gN$ from $G$ to $G/N$.
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\end{theorem}
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