By definition of the classes, this implies that Q must form a “complete” class,
in contradiction to the supposition. Thus, (18.110) holds.
In Sect. 16.3, we described several characteristics of an invariant subgroup.
As readily seen from (16.11) and (18.111), any invariant subgroup consists of
two or more entire classes. Conversely, if a group comprises entire classes, it
must be an invariant subgroup.
(d) Let us think of a product of classes. Let K i and K j be sets described as (18.105).
We define a product K i K j as a set containing products
A
i
ð Þ
α A
j
ð Þ
β
1 α k i , 1 β k j
À
Á
. That is
K i K j ¼
X k i
l¼1
X k j
m¼1
A
i
ð Þ
l A
j
ð Þ
m :
ð18:115Þ
Multiplying (15.115) by g (
8 g 2 ℊ) from the left and by g
À1 from the right, we
get
gK i K j g
À1
¼ gK i g
ð
Þ g
À1 K j g
À1
À
Á ¼ K i K j :
ð18:116Þ
From the above discussion, we get
K i K j ¼
X
l
c ijl K l ,
ð18:117Þ
where c ijl is a positive integer or zero. In fact, when we take gK i K j g
À1 , we
merely permute the terms in (18.117).
(e) If two group elements of the group ℊ ¼ {g 1 , g 2 , Á Á Á, g n } are conjugate to each
other, their inverse elements are conjugate to each other as well. In fact, suppose
that for g μ , g ν 2 K i , we have
g μ ¼ gg ν g
À1
:
ð18:118Þ
Then, taking the inverse of (18.118), we get
g μ
À1
¼ gg ν
À1 g
À1
:
ð18:119Þ
Thus, given a class K i , there exists another class K i
0 that consists of the
inverses of the elements of K i . If g μ 6 ¼ g ν , g μ
À1 6 ¼ g ν
À1 . Therefore, K i and K i
0 are
of the same order. Suppose that the number of elements contained in K i is k i and
that in K i
0 is k i
0 . Then, we have
k i ¼ k i
0 :
ð18:120Þ
706
18 Representation Theory of Groups