Groups, Subgroups and Cosets

Subgroups

Let G be a group and let H be a subset of the elements of G. Now H is a subgroup of G if H is a group, in and of itself, using the same group operator as G. For any positive n, the multiples of n form a subgroup of the integers. If n > 1, the subgroup is a proper subgroup.

If H is a nonempty subset of the group G, H is a subgroup iff every a and b in H implies a/b in H. The forward direction is obvious, so assume the latter. Since H contains a/a, it includes the identity. Since H includes 1/a, we have inverses. Finally H contains ab, via a/(1/b), hence * does not map H to anything outside of H.

Cosets

If you haven't seen cosets before, group theory can be a difficult place to start. (It's so abstract.) Let me explain the idea of a coset in more general terms.

Let H be a substructure inside G. Shifted copies of H are called cosets of H, and they cover all of G without overlap. For example, let G be the xy plane and let H be the x axis. For every real number c, the line running parallel to the x axis, through y = c, is a coset of the x axis. These cosets cover the entire plane without overlap.

The quotient G/H is the collection of cosets. In this example, the quotient is really the y axis.

With this in mind, let's build cosets of the subgroup H inside the group G.

 Given a subgroup H, the right coset of an element x is the set of elements H*x. Left cosets are defined similarly, i.e. x*H. (Note, this convention is reversed for left modules and left ideals, an annoying inconsistency that causes confusion for anyone who works with both nonabelian groups and noncommutative rings.) If a and b are in H, and ax = bx, then cancel x, and a = b. All the elements in a coset are distinct. There are |H| of them. In fact the elements of any given coset, left or right, correspond 1-1 with the elements of H. Thanks to the identity in H, x is always in the coset of x (reflexivity). Since we can always multiply by the inverse of an element in H, x is in the coset of y iff y is in the coset of x (symmetry). If z is in the coset of y is in the coset of x, z is in the coset of x (transitivity). In other words, y = ax and z = by, hence z = bax. We have an equivalence relation, and G can be partitioned into well defined cosets. These cosets have the same size, or cardinality, namely |H|.

Lagrange's Theorem

Lagrange's theorem is an immediate consequence of the above: |H| divides |G| whenever |G| is finite. If |G| = 24, you don't have to look for a subgroup of size 7. There isn't one.

Representative

Sometimes we use designated elements from these cosets to represent them, i.e. coset representatives, somewhat like sending a representative from your district to congress. For brevity, such a representative is called a cosrep. As the name suggests, cosreps are usually taken directly from G, a true representative of the coset, but sometimes we rename the cosets, and assign them different symbols. The group may have no numbers in it at all, but if the cosets behave like integers, we might rename them 1 2 3 … n.

Normal Subgroup

 Let H be a subgroup of G. If x*H = H*x for every x in G, then H is a normal subgroup. In other words, the left and right cosets coincide. This does not mean x commutes with the elements of H, it only means x*H and H*x produce the same set. Of course, if H does commute with every x in G, then H has to be normal. Every subgroup of an abelian group is normal. An equivalent definition says H is normal iff x*y/x is in H for every x in G and every y in H. If left and right cosets are the same, then xy = zx for some z in H, hence xy/x = z, a member of H. Conversly, if xy/x is always some z in H, then xy is the same as zx, a member of the right coset. We can run the other direction by replacing x with x inverse. Multiply by x on the left, giving yx = xz. Each member of the right coset is a member of the left coset, and H is normal. A simple group has no normal subgroups, other than 1 and itself. This may remind you of the definition of a prime number. If G has no proper subgroups at all, it has no normal subgroups, and is simple. Zp, for p prime, is an example, having no subgroups by Lagrange's theorem. An abelian group must be free of subgroups to be simple.