# 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.