18 lines
595 B
TeX
18 lines
595 B
TeX
\section{Elementary Properties of Groups}
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\begin{theorem}[Uniqueness of the Identity]
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In a group $G$, there is only one identity element.
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\end{theorem}
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\begin{theorem}[Cancellation]
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In a group $G$, the right and left cancellation laws hold; that is, $ba = ca$ implies $b = c$, and $ab = ac$ implies $b = c$.
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\end{theorem}
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\begin{theorem}[Uniqueness of Inverses]
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For each element $a$ in a group $G$, there is a unique element $b$ in $G$ such that $ab = ba = e$.
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\end{theorem}
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\begin{theorem}[Socks-Shoes Property]
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For group elements $a$ and $b$, $(ab)^{-1} = b^{-1}a^{-1}$.
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\end{theorem}
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