Decomposition with Respect to Two Subgroups

Earlier we discovered that we could decompose a group G into (right or left) cosets with respect to a subgroup H. In this section we will generalize this decomposition to a decomposition of G with respect to two subgroups, H and K. The decomposition derived in this section was first discovered by Frobenius at the end of the nineteenth century and will be very useful in our discussion of the Sylow theorems. Throughout this section, let G be a group and let H and K be subgroups of G.

Let us define an equivalence relation ~ on G as follows: Say that g ~ g' if and only if g = hg'k for some h H, k K. Let us first prove that ~ is an equivalence relation:

~ is reflexive: g = 1 · g · 1 g ~ g.

~ is symmetric: g ~ g' g = hg'k

(h H, k K) g' = h^-1gk^-1 g' ~ g.

~ is transitive: g ~ g', g' ~ g" g = hg'k, g' = h'g"k'

(h,h' H; k K) g = (hh')g"(k'k) g ~ g".

Thus, ~ is an equivalence relation on G and therefore decomposes G into equivalence classes C_i(i I), where I is some set which we use to label the equivalence classes. If g_i C_i, then C_i consists of those elements of G which are equivalent to g_i, so that

Thus,

G = Hg_iK.

Moreover, since C_iC_j = if ij, we have

Hg_iK Hg_jK =    (ij)

We will be interested in the case where G is a finite group of order n. Suppose that there are t equivalence classes C₁,...,C_t. Then (1) may be written

G = Hg₁K ... Hg_tK.

Therefore, by (2) and (3),

Now it is clear that

since if Hg_iK = {x₁,...,x_a}, then g_i^-1Hg_iK = {g_i^-1x₁,...,g_i^-1x_a}. Let H_i = g_i^-1Hg_i. Then H_i < G and |H_i| = |H|. Moreover, by (5),

By equation (4) of the section on isomorphism theorems, we see that

|H_iK| = = .

Therefore, by (6) and (7), we have

|Hg_iK| = .

By combining (4) and (8), we may summarize our results as follows:

Theorem 1 (Frobenius): Let G be a finite group of order n, let H and K be subgroups of order p and q, respectively, and let g₁,...,g_t be representatives of the equivalence classes with respect to ~. Then

n =

where u_i = |g_i^-1Hg_i K|.

If in Theorem 1, we set H = {1}, then we see that p = 1 and u_i = 1 for all i, so that n = qt, which is Lagrange's theorem, since the equivalence classes C_i are just cosets g_iK.