68 lines
1.8 KiB
TeX
68 lines
1.8 KiB
TeX
\section{Isomorphisms}
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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 while preserving multiplication structure. This mapping is called an \textit{isomorphism}.\\
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\vspace{2mm}
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This means that if groups $A$ and $B$ are isomorphic under $f$, \\
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$a_1 \ast a_2 = a_3$ in A implies that $f(a_1) \ast f(a_2) = f(a_3)$ in B.
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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 & 3 & 4 \\
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2 & 2 & 4 & 1 & 3 \\
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3 & 3 & 1 & 4 & 2 \\
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4 & 4 & 3 & 2 & 1 \\
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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 under $f$. 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 under $f$. \\
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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 under $f$. 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{}<howmanygroups>
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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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\begin{solution}
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There is only one nonisomorphic two-element group. \\
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The same is true of a three-element group. \\
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See \texttt{https://oeis.org/A000001}, titled \say{Number of groups of order n}
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\end{solution}
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\vfill
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\problem{}
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Show that the groups $(\mathbb{R}, +)$ and $(\mathbb{R}^+, \times)$ are isomorphic.
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\vfill
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\pagebreak |