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.

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Parents:

  • Group

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