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We shall now pass on to groups of finite order. It is clear that here we must have to do with many properties which have no direct analogues in the theory of continuous groups or in that of discontinuous groups in general; those properties, namely, which depend on the fact that the number of distinct operations in the group is finite.
Every group of finite order N can therefore be represented in concrete form as a transitive group of permutations on N symbols.
Properties of a group which depend on the order.
Sylow's theorem.
An Abelian group contains subgroups whose orders are any given factors of the order of the group. In fact, since every subgroup H of an Abelian group G and the corresponding factor groups G/H are Abelian, this result follows immediately by an induction from the case in which the order contains n prime factors to that in which it contains n + 1. For a group which is not Abelian no general law can be stated as to the existence or non-existence of a subgroup whose order is an arbitrarily assigned factor of the order of the group. In this connexion the most important general result, which is independent of any supposition as to the order of the group, is known as Sylow's theorem, which states that if p^a is the highest power of a prime p which divides the order of a group G, then G contains a single conjugate set of subgroups of order p^a, the number in the set being of the form 1 + kp. Sylow's theorem may be extended to show that if p^a? is a factor of the order of a group, the number of subgroups of order p^a? is of the form 1 + kp. If, however, p^a? is not the highest power of p which divides the order, these groups do not in general form a single conjugate set.
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.
Composition-series of a group.
Isomorphism of a group with itself.
Permutation-groups.
Groups of linear substitutions.
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 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.
A group of linear substitutions on m variables is said to be "reducible" when it is possible to choose m? linear functions of the variables which are transformed among themselves by every substitution of the group. When this cannot be done the group is called "irreducible." It can be shown that a group of linear substitutions, of finite order, is always either irreducible, or such that the variables, when suitably chosen, may be divided into sets, each set being irreducibly transformed among themselves. This being so, it is clear that when the irreducible representations of a group of finite order are known, all representations may be built up.
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