Alternating group

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last updated 18th Jun 2016

The alternating group 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).

$A_{n}$ 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

The alternating group $A_{n}$ is of 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.

$A_{n}$ is generated by its $3$ -cycles. (Proof.)

$A_{n}$ is simple. (Proof.)

The conjugacy classes of $A_{n}$ are easily characterised.

Parents:

Group

Children:

The alternating groups on more than four letters are simple

The collection of even-signed permutations is a group

and 4 more