# Group coset

Given a sub­group $$H$$ of group $$G$$, the left cosets of $$H$$ in $$G$$ are sets of the form $$\{ gh : h \in H \}$$, for some $$g \in G$$. This is writ­ten $$gH$$ as a short­hand.

Similarly, the right cosets are the sets of the form $$Hg = \{ hg: h \in H \}$$.

# Examples

knows-req­ui­site(Sym­met­ric group):

## Sym­met­ric group

In $$S_3$$, the sym­met­ric group on three el­e­ments, we can list the el­e­ments as $$\{ e, (123), (132), (12), (13), (23) \}$$, us­ing cy­cle no­ta­tion. Define $$A_3$$ (which hap­pens to have a name: the al­ter­nat­ing group) to be the sub­group with el­e­ments $$\{ e, (123), (132) \}$$.

Then the coset $$(12) A_3$$ has el­e­ments $$\{ (12), (12)(123), (12)(132) \}$$, which is sim­plified to $$\{ (12), (23), (13) \}$$.

The coset $$(123)A_3$$ is sim­ply $$A_3$$, be­cause $$A_3$$ is a sub­group so is closed un­der the group op­er­a­tion.$$(123)$$ is already in $$A_3$$. <div>

more ex­am­ples, with differ­ent requirements

# Properties

• The left cosets of $$H$$ in $$G$$ par­ti­tion $$G$$. (Proof.)

• For any pair of left cosets of $$H$$, there is a bi­jec­tion be­tween them; that is, all the cosets are all the same size. (Proof.)

# Why are we in­ter­ested in cosets?

Un­der cer­tain con­di­tions (namely that the sub­group $$H$$ must be nor­mal), we may define the quo­tient group, a very im­por­tant con­cept; see the page on “left cosets par­ti­tion the par­ent group” for a glance at why this is use­ful. there must be a less clumsy way to do it

Ad­di­tion­ally, there is a key the­o­rem whose usual proof con­sid­ers cosets (La­grange’s the­o­rem) which strongly re­stricts the pos­si­ble sizes of sub­groups of $$G$$, and which it­self is enough to clas­sify all the groups of or­der $$p$$ for $$p$$ prime. La­grange’s the­o­rem also has very com­mon ap­pli­ca­tions in num­ber the­ory, in the form of the Fer­mat-Euler the­o­rem.

Children:

Parents:

• Group

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