Let be a group and a subgroup of . Then for each element we can define a *left coset* of by

.

That is, is the set we get by combining (on the left) with every element of . For example, given the subgroup (this was the other subgroup of —did you find it?), the left coset corresponding to is . A few observations:

- The cosets corresponding to different elements of might be the same. For example, the left coset of corresponding to is , just like the coset for .
- Can you find the other possible (left) cosets of in ? What do you notice?
- As you may guess, there are also things called
*right cosets*, denoted , where we combine with an element on the*right*. is not such a good example anymore, since in the binary operation is commutative, that is, , so left and right cosets are the same thing. In general, though, the binary operation of a group does not have to be commutative. - There is nothing special about
*left*cosets as opposed to*right*cosets. In our proof of Lagrange’s Theorem we will use left cosets, but we could equally well replace all the left cosets by right cosets (and flip a few other things around) to get a different but equally valid proof. - As an interesting aside, when the left and right cosets of a subgroup coincide, we say that the subgroup is
*normal*. (Hence every subgroup of a group with a commutative binary operation is normal; but this can also happen even when the binary operation is not commutative.) It turns out that these normal subgroups are very important. Normal subgroups of a group are kind of like the divisors of an integer; you can “divide” a group by one of its normal subgroups to get a “quotient group”. And yes, there are special groups called*simple*groups which don’t have any normal subgroups, and are kind of like prime numbers—there is a suitable sense in which every finite group can be uniquely decomposed into a “product” of simple groups, just like integers can be uniquely decomposed into a product of prime factors. But this is getting way off on a tangent! (I told you this proof would hint at some very cool, deeper group theory.)

Just one more observation for today. For any , consider the function which combines its input with (on the left). This function is injective, that is, one-to-one: if , then by definition , and combining both sides with on the left, , hence . (In a group we can always cancel things from both sides of an equation—though only from the end! For example, from it does *not* follow that .) Conversely, this means that if , then .

When we form the left coset , we are applying the function to every element of . The fact that this function is injective means it can’t “collapse” multiple elements of into the same element in the result. This shows that the coset has to have the *same size as* : there is exactly one element in for each element of , and they all have to be different.

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