The importance of Sylow’s theorem in discussing the structure of a group of given order need hardly be insisted on. Thus, as a very simple instance, a group whose order is the product p1p2 of two primes (p1 < p2) must have a self-conjugate subgroup of order p2, since the order of the group contains no factor, other than unity, of the form 1 + kp2. The same again is true for a group of order p12p2, unless p1 = 2, and p2 = 3.
There is one other numerical property of a group connected with its order which is quite general. If N is the order of G, and n a factor of N, the number of operations of G, whose orders are equal to or are factors of n, is a multiple of n.
As already defined, a composite group is a group which contains one or more self-conjugate subgroups, whose orders are greater than unity. If H is a self-conjugate subgroup of G, the factor-group Composition-series of a group. G/H may be either simple or composite. In the former case G can contain no self-conjugate subgroup K, which itself contains H; for if it did K/H would be a self-conjugate subgroup of G/H. When G/H is simple, H is said to be a maximum self-conjugate subgroup of G. Suppose now that G being a given composite group, G, G1, G2, ..., Gn, 1 is a series of subgroups of G, such that each is a maximum self-conjugate subgroup of the preceding; the last term of the series consisting of the identical operation only. Such a series is called a composition-series of G. In general it is not unique, since a group may have two or more maximum self-conjugate subgroups. A composition-series of a group, however it may be chosen, has the property that the number of terms of which it consists is always the same, while the factor-groups G/G1, G1/G2, ..., Gn differ only in the sequence in which they occur. It should be noticed that though a group defines uniquely the set of factor-groups that occur in its composition-series, the set of factor-groups do not conversely in general define a single type of group. When the orders of all the factor-groups are primes the group is said to be soluble.
If the series of subgroups G, H, K, ..., L, 1 is chosen so that each is the greatest self-conjugate subgroup of G contained in the previous one, the series is called a chief composition-series of G. All such series derived from a given group may be shown to consist of the same number of terms, and to give rise to the same set of factor-groups, except as regards sequence. The factor-groups of such a series will not, however, necessarily be simple groups. From any chief composition-series a composition-series may be formed by interpolating between any two terms H and K of the series for which H/K is not a simple group, a number of terms h1, h2, ..., hr; and it may be shown that the factor-groups H/h1, h1/h2, ..., hr/K are all simply isomorphic with each other.
A group may be represented as isomorphic with itself by transforming all its operations by any one of them. In fact, if SpSq = Sr, then S−1SpS·S−1SqS = S−1SrS. An isomorphism of the Isomorphism of a group with itself. group with itself, established in this way, is called an inner isomorphism. It may be regarded as an operation carried out on the symbols of the operations, being indeed a permutation performed on these symbols. The totality of these operations clearly constitutes a group isomorphic with the given group, and this group is called the group of inner isomorphisms. A group is simply or multiply isomorphic with its group of inner isomorphisms according as it does not or does contain self-conjugate operations other than identity. It may be possible to establish a correspondence between the operations of a group other than those given by the inner isomorphisms, such that if S′ is the operation corresponding to S, then S′pS′q = S′r is a consequence of SpSq = Sr. The substitution on the symbols of the operations of a group resulting from such a correspondence is called an outer isomorphism. The totality of the isomorphisms of both kinds constitutes the group of isomorphisms of the given group, and within this the group of inner isomorphisms is a self-conjugate subgroup. Every set of conjugate operations of a group is necessarily transformed into itself by an inner isomorphism, but two or more sets may be interchanged by an outer isomorphism.
A subgroup of a group G, which is transformed into itself by every isomorphism of G, is called a characteristic subgroup. A series of groups G, G1, G2, ..., 1, such that each is a maximum characteristic subgroup of G contained in the preceding, may be shown to have the same invariant properties as the subgroups of a composition series. A group which has no characteristic subgroup must be either a simple group or the direct product of a number of simply isomorphic simple groups.
It has been seen that every group of finite order can be represented as a group of permutations performed on a set of symbols whose number is equal to the order of the group. In general such Permutation-groups. a representation is possible with a smaller number of symbols. Let H be a subgroup of G, and let the operations of G be divided, in respect of H, into the sets H, S2H, S3H, ..., SmH. If S is any operation of G, the sets SH, SS2H, SS3H, ..., SSmH differ from the previous sets only in the sequence in which they occur. In fact, if SSp belong to the set SqH, then since H is a group, the set SSpH is identical with the set SqH. Hence, to each operation S of the group will correspond a permutation performed on the symbols of the m sets, and to the product of two operations corresponds the product of the two analogous permutations. The set of permutations, therefore, forms a group isomorphic with the given group. Moreover, the isomorphism is simple unless for one or more operations, other than identity, the sets all remain unaltered. This can only be the case for S, when every operation conjugate to S belongs to H. In this case H would contain a self-conjugate subgroup, and the isomorphism is multiple.
The fact that every group of finite order can be represented, generally in several ways, as a group of permutations, gives special importance to such groups. The number of symbols involved in such a representation is called the degree of the group. In accordance with the general definitions already given, a permutation-group is called transitive or intransitive according as it does or does not contain permutations changing any one of the symbols into any other. It is called imprimitive or primitive according as the symbols can or cannot be arranged in sets, such that every permutation of the group changes the symbols of any one set either among themselves or into the symbols of another set. When a group is imprimitive the number of symbols in each set must clearly be the same.
The total number of permutations that can be performed on n symbols is n!, and these necessarily constitute a group. It is known as the symmetric group of degree n, the only rational functions of the symbols which are unaltered by all possible permutations being the symmetric functions. When any permutation is carried out on the product of the n(n − 1)/2, differences of the n symbols, it must either remain unaltered or its sign must be changed. Those permutations which leave the product unaltered constitute a group of order n!/2, which is called the alternating group of degree n; it is a self-conjugate subgroup of the symmetric group. Except when n = 4 the alternating group is a simple group. A group of degree n, which is not contained in the alternating group, must necessarily have a self-conjugate subgroup of index 2, consisting of those of its permutations which belong to the alternating group.
Among the various concrete forms in which a group of finite order can be presented the most important is that of a group of linear Groups of linear substitutions. substitutions. Such groups have already been referred to in connexion with discontinuous groups. Here the number of distinct substitutions is necessarily finite; and to each operation S of a group G of finite order there will correspond a linear substitution s, viz.