## Double Cosets

**Point of Post: **In this post we discuss the notion of double cosets in a group .

*Motivation*

There is a natural extension of the notion of cosets which are double cosets, namely cosets on ‘both’ sides. While of algebraic interest in their own right we shall see that they are mostly of interest in other fields such as ring theory and representation theory.

*Double Cosets*

Let be a group and . Consider the mapping of the direct product times to given by left multiplication by and right inverse multiplication by . In other words given by . We claim that is an action of on . Indeed, it’s clear that for every and for every and we have

Thus, is a -action on as claimed. Let be an orbit of . It’s evident that . We call such an orbit a *double coset of and . *We denote the set of all double cosets of and in as

We can think about double cosets in another way. Namely, let act on by the usual right inverse multiplication (or not inverse if you dont’ care about right actions). Consider then the action of on given by left multiplication. Clearly the double cosets and this action are very related. Indeed:

**Theorem: ***Let be a group and . Then, if is a double coset of and then*

It’s evident that is contained in . To see the converse we note that if then but by definition we have that and so and so evidently is in . To see that they are disjoint we merely recall that the set of right cosets partition .

Of course there is an analogous s result by considering the obvious -action on the set of all right cosets of in .

We can now use to find the cardinality of . Namely:

**Theorem: ***Let be a finite group and . Then, for any double coset .*

**Proof: **We have by the previous theorem that

But, by the orbit-stabilizer theorem we have that . What we claim though is that . Indeed, if then for some and so and thus . Conversely, if then and so that there exists some so that and so so that . Thus, as claimed and so the theorem follows.

Applying the exact same technique considering instead the -action on we see that

**Theorem: ***Let be a finite group and . Then, for any double coset one has*

**References:**

1. Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Prin**t
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[…] It is a commonly used theorem in finite group theory that if is a finite group and such that is the smallest prime dividing then . We have already seen a proof of this fact by considering the homomorphism which is the induced map from acting on by left multiplication, and proving that . We now give an even shorter (and the just mentioned proof is already short) proof of this fact using double cosets. […]

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