# Alternating group

https://arbital.com/p/alternating_group

by Patrick Stevens Jun 17 2016 updated Jun 18 2016

The alternating group is the only normal subgroup of the symmetric group (on five or more generators).

The alternating group $A_n$ is defined as a certain subgroup of the Symmetric group $S_n$: namely, the collection of all elements which can be made by multiplying together an even number of transpositions. This is a well-defined notion (proof).

%%%knows-requisite(Normal subgroup): $A_n$ is a Normal subgroup of $S_n$; it is the quotient of $S_n$ by the sign homomorphism. %%%

# Examples

• A cycle of even length is an odd permutation in the sense that it can only be made by multiplying an odd number of transpositions. For example, $(132)$ is equal to $(13)(23)$.

• A cycle of odd length is an even permutation, in that it can only be made by multiplying an even number of transpositions. For example, $(1354)$ is equal to $(54)(34)(14)$.

• The alternating group $A_4$ consists precisely of twelve elements: the identity, $(12)(34)$, $(13)(24)$, $(14)(23)$, $(123)$, $(124)$, $(134)$, $(234)$, $(132)$, $(143)$, $(142)$, $(243)$.

# Properties

%%%knows-requisite(Normal subgroup): The alternating group $A_n$ is of [index_of_a_subgroup index] $2$ in $S_n$. Therefore $A_n$ is normal in $S_n$ (proof). Alternatively we may give the homomorphism explicitly of which $A_n$ is the kernel: it is the sign homomorphism. %%%