# Alternating group

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 $$2$$ 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.

Children:

Parents:

• Group

The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.