56 lines
1.4 KiB
TeX
56 lines
1.4 KiB
TeX
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\section{Isomorphism}
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\definition{}
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We say two groups are \textit{isomorphic} if we can create a bijective mapping between them.
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\problem{}
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Recall your tables from \ref{modtables}: \\
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\begin{center}
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\begin{tabular}{c | c c c c}
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+ & 0 & 1 & 2 & 3 \\
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\hline
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0 & 0 & 1 & 2 & 3 \\
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1 & 1 & 2 & 3 & 0 \\
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2 & 2 & 3 & 0 & 1 \\
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3 & 3 & 0 & 1 & 2 \\
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\end{tabular}
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\hspace{1cm}
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\begin{tabular}{c | c c c c}
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\times & 1 & 2 & 3 & 4 \\
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\hline
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1 & 1 & 2 & 4 & 3 \\
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2 & 2 & 4 & 3 & 1 \\
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3 & 4 & 3 & 1 & 2 \\
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4 & 3 & 1 & 2 & 4 \\
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\end{tabular}
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\end{center}
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Are $(\mathbb{Z}/4, +)$ and $( (\mathbb{Z}/5)^\times, \times)$ isomorphic? If they are, find a bijection that maps one to the other.
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\vfill
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\problem{}
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Let groups $A$ and $B$ be isomorphic, and let $f: A \to B$ be a bijection. Show that $f(e_A) = e_B$, where $e_A$ and $e_B$ are the identities of $A$ and $B$.
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\vfill
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\problem{}
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Let groups $A$ and $B$ be isomorphic, and let $f: A \to B$ be a bijection. \\
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Show that $f(a^{-1}) = f(a)^{-1}$ for all $a \in A$.
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\vfill
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\problem{}
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Let groups $A$ and $B$ be isomorphic, and let $f: A \to B$. Show that $f(a)$ and $a$ have the same order.
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\vfill
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\pagebreak
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\problem{}
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Find all distinct groups of two elements. \\
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Find all distinct groups of three elements. \\
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Groups that are isomorphic are not distinct.
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\vfill
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\problem{}
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Show that the groups $(\mathbb{R}, +)$ and $(\mathbb{Z}^+, \times)$ are isomorphic.
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\vfill
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\pagebreak
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