---- > [!definition] Definition. ([[alternating group]]) > The [[subgroup]] of the [[symmetric group]] given by the set of [[parity of a permutation|even permutations]] on $n$ letters is called the **alternating group of degree $n$**. It is denoted $A_{n}$. > [!justification] > Let's show this is indeed a [[subgroup]]. Clearly $e_{S_{n}} \in A_{n}$. If $\sigma$ is an [[parity of a permutation|even]] [[permutation]], then $\sigma$ may be decomposed into $2k$ transpositions. Likewise an [[parity of a permutation|even permutation]] $\tau$ can be decomposed into $2\ell$ transpositions. Hence $\sigma \tau$ consists of $2(n+k)$ transpositions so $\sigma \tau \in A_{n}$. Inverses are also in $A_{n}$, since undoing an even number of transpositions is accomplished via an even number of transpositions. > [!basicproperties] > - $|A_{n}|= \frac{n!}{2}=\frac{|S_{n}|}{2}$. > [!proof] Proof of Basic Properties. > - See [[order of quotient group is quotient of orders]] > [!basicexample] Example. (The structure of $A_{4}$) $A_{4}$ is the [[group]] of [[parity of a permutation|even permutations]] on $4$ letters. $2$-cycles and $4$-cycles are odd, $3$-cycles are even. Thus the elements of $A_{4}$ are >- The $3$-cycles of $S_{4}$: $(123), (124), (132), (134), (142), (143), (234), (243)$ >- The disjoint products of $2$-cycles in $S_{4}$: $(12)(34), (13)(24), (14)(23)$ > - $(e)$. ^528873 ---- #### ---- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM outgoing([[]]) FLATTEN file.tags GROUP BY file.tags as Tag