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Alternating groups

We are now going to introduce an important subgroup of the symmetric group $S_n$. First we introduce another way of writing permutations.


\begin{thm}
Every permutation can be written as a product of $2$-cycles
(also called {\em transpositions}).
\end{thm}


\begin{proof}
We already know that a permutation can be written as a product
of ...
...i_3)\ldots (i_{k-1}\ i_k);
\end{displaymath}this proves the theorem.
\end{proof}

A decomposition of a permutation into a product of transpositions is by no means unique; for instance

\begin{displaymath}
(2\ 3)=(1\ 2)(2\ 3)(1\ 3).
\end{displaymath}

However, the parity of the number of transpositions in any decomposition of a given permutation does not change.


\begin{de}
A permutation is {\em even} (respectively {\em odd})
if it can be written as a product of an even
(respectively odd) number of transpositions.
\end{de}


\begin{thm}
A permutation cannot be both even and odd.
\end{thm}


\begin{proof}
Consider the polynomial of $n$\ variables:
\begin{displaymath}
P=P...
...ts(\delta_{2r+1}(P)))=-P,
\end{displaymath}which is a contradiction.
\end{proof}


\begin{exc}
% latex2html id marker 2107A $k$-cycle is even if and only if $k$\ is odd! Hint: see the
proof of Theorem \ref{thm34}.
\end{exc}


\begin{thm}
The set $A_n$\ of all even permutations in $S_n$\ is a subgroup of $S_n$of order $n!/2$.
\end{thm}


\begin{proof}
% latex2html id marker 2112For $\sigma=\gamma_1\ldots\gamma_{2k}...
...rt+\vert O_n\vert=2\vert A_n\vert$,
and hence $\vert A_n\vert=n!/2$.
\end{proof}


\begin{de}
$A_n$\ is called the {\em alternating group} on $\{1,\ldots,n\}$.
\end{de}


\begin{exc}
List the elements of $A_4$.
\end{exc}


\begin{exc}
Prove that $O_n$\ is not a subgroup of $S_n$.
\end{exc}



Edmund F Robertson

11 September 2006