It follows from the definition of a group that it must always be possible to choose from its operations a set such that every operation of the group can be obtained by combining the operations of the set and their inverses. If the set is such that no one of the operations belonging to it can be represented in terms of the others, it is called a set of _independent generating_ operations. Such a set of generating operations may be either finite or infinite in number. If A, B, ..., E are the generating operations of a group, the group generated by them is represented by the symbol {A, B, ..., E}. An obvious extension of this symbol is used such that {A, H} represents the group generated by combining an operation A with every operation of a group H; {H1, H2} represents the group obtained by combining in all possible ways the operations of the groups H1 and H2; and so on. The independent generating operations of a group may be subject to certain relations connecting them, but these must be such that it is impossible by combining them to obtain a relation expressing one operation in terms of the others. For instance, AB = BA is a relation conditioning the group {A, B}; it does not, however, enable A to be expressed in terms of B, so that A and B are independent generating operations. Transitivity and primitivity. Let O, O', O", ... be a set of objects which are interchanged among themselves by the operations of a group G, so that if S is any operation of the group, and O any one of the objects, then O.S is an object occurring in the set. If it is possible to find an operation S of the group such that O.S is any assigned one of the set of objects, the group is called _transitive_ in respect of this set of objects. When this is not possible the group is called _intransitive_ in respect of the set. If it is possible to find S so that any arbitrarily chosen n objects of the set, O1, O2, ..., O_n are changed by S into O'1, O'2, ..., O'n respectively, the latter being also arbitrarily chosen, the group is said to be n-ply transitive. If O, O', O", ... is a set of objects in respect of which a group G is transitive, it may be possible to divide the set into a number of subsets, no two of which contain a common object, such that every operation of the group either interchanges the objects of a subset among themselves, or changes them all into the objects of some other subset. When this is the case the group is called _imprimitive_ in respect of the set; otherwise the group is called _primitive_. A group which is doubly-transitive, in respect of a set of objects, obviously cannot be imprimitive. Illustrations of the group idea.