PROPERTIES INDEPENDENT OF THE MODE OF
REPRESENTATION OF A GROUP If N be the order of a group G and r the order of a sub-group ' H, then r is an integral divisor of N. For, let the subgroup be H =
= I,
,
and let
be any element of G not found in H. The elements of the set hig2, h282, h3g2, ,
are all distinct by (3a), and none are found in H. Proceeding in this manner, the elements of G can be arranged in the form of a rectangle whose dimensions are r by p. In this rectangle no ele ment is repeated for if
=
j), then gi = hr
is an element already enumerated. It follows that N= pr. The integer p = N is called the index of H under G. This theorem, due to r Lagrange, is the chief cornerstone of the theory of finite groups.
Hgi and is called a right-hand co-set of G with respect to H. The group G may then be indicated by the notation G=H+Hg2-1-Hg3+ ... +HgP.
G can be expressed in terms of left-hand co-sets so that G= H+g2'H+go'H+
where
g3', , gp' are suitably chosen. In particular, the group may be expresses in the Corm G = H-Fg2 iti+g3 'H+ . . .
where the elements of
are the reciprocals of the elements of Hgi.
G in terms of double co-sets Rgi H. Thus G = RH+
Invariant Sub-groups.-An element
where h is an element of a group H and g is either an element of H, or of a larger group G containing H, is called a transform of h, or a conjugate of h. The set of elements obtained by transforming all the elements of H form a group H'. For if
and
be two elements of H, the product g ihig g
= g
is again in the set of transforms. The group H' is called a conjugate of H and is simply isomorphic to it. We may write H' = g iHg.
H of a group G be transformed in succession by all the elements of G, a set of groups H, H', H", , all con jugate to H is obtained. The group
composed of the elements common to the conjugate groups H, H', H", , has the im portant property that g
=
for every element g of G. Such a group is called a self-conjugate subgroup, or an invariant subgroup, or a normal divisor, of G. The alternating group G,,, is an invariant subgroup of the symmetric group G,,,.
group G which contains no invariant sub group except the trivial cases of the identity and the group itself is called a simple group; otherwise it is composite. Every group of prime order is simple. A fundamental theorem is that the alternating group on n letters is simple except for the single case n= 4. The icosahedral group which is simply isomorphic with the alternating group on 5 letters, is the smallest simple group of composite order. The orders of the remaining simple groups of composite order less than 2000 are 168, 360, 504, 66o, 1o92.
The number of simple groups of composite order is infinite.
Dickson in his Linear Groups with an Exposition of the Galois Field Theory has enumerated 78 known simple groups of com posite orders less than one billion. Of the 53 whose orders are less than one million, all but 3 belong to known infinite systems of simple groups. There are two distinct simple groups of order 20,160. Dickson has shown further that there is an infinite number of orders corresponding to which there exist more than one type of simple group. It is known that a group cannot be simple if its order N has one of the forms p, page, pqr or
r, )distinct primes; or if it has fewer than 6 prime factors (N> 2000); or if it is not divisible by one of the numbers 12, 16, 56 (N= 2n). So far no simple group of odd composite order has been discovered.
be an invariant subgroup of a group the co-sets of G with respect to
may be looked upon as the elements of a group. The product
Gigk is interpreted to mean the totality of distinct elements of the form
as g and g' run through the elements of
Since
is invariant in G,
= Gigi and
=
It follows that Gigi G1gi = Gi giGi g= GiGigigi =
The product of two co-sets is a co-set and the co-sets form a group I' =
,
when and only when
is invariant. This group is called the quotient, or factor group, or the complementary group of
with respect to G and is usually written G The co-set
is the identical element and the G1 element
is inverse to
Between the groups G and G G1 there is an
I)-fold isomorphism.
invariant subgroup
of G which is contained in no other invariant subgroup of G is called a maximal invariant subgroup of G. The series G,
G2, , 1,
terminating with the identity group, in which every group is a maximal invariant subgroup of the preceding group is called a series of composition of G. The integers
defined to denote the respective indices of
G2, under the preceding group are called the factors of composition of G. The factor G2 groups G ' - , - , are all simple. For if
had an in
G2 Go Gs+1 variant subgroup, Gi would contain a proper invariant subgroup
which would contain
If the factors of composition are all prime numbers, the group is said to be solvable, a term carried over from the theory of equations (see EQUATIONS, THEORY OF), where it is shown that the necessary and sufficient condition that an equation be solva ble by radicals is that its group have only prime numbers for its factors of composition. If a group has more than one series of composition, the theorem of Camille Jordan published in his Traite des Substitutions in 187o, asserts that the factors of corn position are, apart from their order, the same for every series of composition. Jordan's theorem is a direct consequence of a more general theorem published in 1889 by Holder proving that the G
G2 factor groups - - - , are identical in some order with G1 G2 G3 the factor groups for any other series of composition.
s be any operation of a group G, the distinct elements
, sh obtained by trans forming s by every element of G is called a complete conjugate set. Clearly, s itself will be found in the set and, moreover, the complete set of transforms of any element si of the set is identical with the complete set obtained from s. If t be any element in the group not in the set determined by s, a new complete set of elements conjugate to t may be found. In this way the elements of the group G may be separated into non-overlapping complete conjugate sets. If hi be the number in a set and r the number of complete conjugate sets, the order N of G may be written N=h1-1-h2+ . .
Since the identical element is transformed into itself by every element at least one h is unity. If the group is Abelian, every h is unity. Every h is a divisor of the order N of the group.
An element which is commutative with every element of the group is called an isolated element. The totality of the isolated elements form an Abelian subgroup called the central of G. For any two elements s and t there is a unique third element c such that st = tsc so that c = s
This element c was called by R. Dedekind the commutator of s and t. The commutators of a group G do not necessarily form a group but they generate a group. This group, which may be identical with G, is an invariant subgroup C of G called the commutator group. It has the property that it is the smallest invariant subgroup of G for which the factor group G/C is Abelian.
knowledge of the structure of finite groups depends largely upon a set of closely related theorems of which Sylow's theorem was the first to be proven. In a paper of fundamental importance published in 1872, L. Sylow showed that if pa is the highest power of a prime p contained in the order of a group G, then G must contain at least one subgroup of order pa and that if there are more than one, they form a conjugate system whose number is I
where k is a positive integer. Such groups are called Sylow subgroups.
It is shown that a group whose order is pa contains sub groups of orders
,
p, whence it follows that G contains subgroups of order pr for r < a. Froebenius extended Sylow's theorem by showing that the number of subgroups of order pr, (r
Abelian group, i.e., a group whose ele ments are all commutative, has the property that there exist within the group a set of elements
A2, A3, of orders a1, a2, a3, such that every element 0 of the group may be expressed in the form
A2"2
,
I, 2, ,
The elements
form a base, the elements of which may be selected in more than one way.
In particular, the elements may be selected so that their orders are powers of the prime factors occurring in the order of the group.
An Abelian group G of order
,
is the direct product of Abelian groups
. - , Gpa of orders
,
respectively. It follows that the problem of determining all Abelian groups of any given order is solved when we know the possible types of Abelian groups whose orders are prime powers.
The distinct types of Abelian groups whose orders are pn are given by the form
P3a3
(s =o, I, 2, 3, .,
I) where
P2, ,
are generators of orders pni, pn2, ... , re,
and
is any partition of n. Thus, there are
three distinct Abelian groups of order
corresponding to the partitions 3, 2 + I, I + I + I, of 3.
The type of an Abelian group of order pn is completely specified by the partition.
G =
= I, S2, S3, ,
be a group and t an element not commutative with every element of G, the conjugate group
r r11
=
S2, Sg, .,SNJ,
where
=
Si t1, (i = I, 2, , N), is simply isomorphic with G. If t be an element of G, then G' is identical with G though its elements do not necessarily occur in the same order. We have, therefore, an isomorphism of the group with itself. If G is Abelian, a correspondence which shows the group to be isomorphic to itself may be set up by making each element correspond to its inverse.
The isomorphism in which each element corresponds to itself is called the identical isomorphism. For a group of order 2 the identical isomorphism is the only one possible. In every other case there exist isomorphisms different from the identical iso morphism.
An isomorphism of a group with itself defines a substitution S' I,
,
S2, S3 , , SN and the totality of the isomorphisms form an intransitive sub stitution group L isomorphic to G. An isomorphism determined by taking the transforms
of all the elements of G is called a cogredient isomorphism. All other isomorphisms of G with itself are called contragredient. An important theorem is that the cogredient isomorphisms of a group G form an invariant sub group of the group of isomorphisms.
