Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

~ 10]

CO-SETS AND DOUBLE CO-SETS

25

It will be proved later (~ 33) that it is always possible to select
the X-1 multipliers in such a manner that t', = ta, = 2,..., X.
By taking the inverses of each of the co-sets in the formula of
the first paragraph of this section, it results that
GH+t2-1H+... +t-lH.
Hence it is also possible to replace t', by tai~ in the preceding
formula.
If H1 and H2 are any two subgroups of G, the symbol
H1taH2
is called a double co-set* of G as regards H1 and H2. It implies
that each of the substitutions of Hta, is multiplied on the right
by every substitution of H2. All of these products are represented by the single symbol HltaH2. While all of the substitutions of a co-set are distinct, those of a double co-set need
not be distinct. If one substitution of G occurs exactly k times
in such a double co-set, every substitution of the double co-set
occurs exactly k times among the products represented by this
double co-set. We proceed to prove this statement.
Consider the product of the two groups H1 H2, that is, all
the products obtained by multiplying every substitution of
H1 on the right by all the substitutions of H2. If H1 and H2
have exactly p substitutions in common, it is clear that each
substitution of H1 and each of H2 will appear exactly p times
in H1 H2. In fact, if ti is any substitution of H1, the p substitutions obtained by multiplying tj on the right by the substitutions which are common to H1 and H2 will yield the same
products when they are multiplied on the right by H2. As
no other substitution of H1 can yield any of these products,
being in the same co-set of G as regards H2, it results that
the product H1iH2 involves each one of its substitutions exactly
* Double co-sets were first used by A. L. Cauchy, Paris Comptes Rendus,
vol. 22 (1846), p. 630. They were more fully developed by Frobenius,
Crelle, vol. 101 (1887), p. 273. The term double co-set was first used with this
meaning in Bulletin of the American Mathematical Society, vol. 17 (1911), p. 292.