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|HK| = If H is a subgroup of G then aH = {a · h | a in G, h in H} is a left coset Ha = {h · a | a in G, h in H} is a right coset

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 element of H, k element of K. Let us first prove that ~ is an equivalence relation:

~ is reflexive: g = 1 · g · 1 implies g ~ g.

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

(h element of H, k element of K) implies g' = h-1gk-1 implies g' ~ g.

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

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

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

Ci = HgiK.

Thus,

(1)
G = HgiK.

Moreover, since CiintersectionCj = empty set if inot equalj, we have

(2)
HgiK intersectionHgjK = empty set   (inot equalj)

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

(3)
G = Hg1K union ... union HgtK.

Therefore, by (2) and (3),

(4)
|G| = |Hg1K| + ... + |HgtK|.

Now it is clear that

(5)
|HgiK| = |gi-1HgiK|,

since if HgiK = {x1,...,xa}, then gi-1HgiK = {gi-1x1,...,gi-1xa}. Let Hi = gi-1Hgi. Then Hi < G and |Hi| = |H|. Moreover, by (5),

(6)
|HgiK| = |HiK|.

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

(7)
|HiK| = = .

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

(8)
|HgiK| = .

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,...,gt be representatives of the equivalence classes with respect to ~. Then

n =

where ui = |gi-1Hgi intersectionK|.

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