A group is spoken of as _simple_ when it has no self-conjugate subgroup other than that constituted by the identical operation alone. A group which has a self-conjugate subgroup is called _composite_. Let G be a group constituted of the operations S, T, U, ..., and g a second group constituted of s, t, u, ..., and suppose that to each operation of G there corresponds a single operation of g in such a way that if ST = U, then _st_ = u, where s, t, u are the operations corresponding to S, T, U respectively. The groups are then said to be _isomorphic_, and the correspondence between their operations is spoken of as an _isomorphism_ between the groups. It is clear that there may be two distinct cases of such isomorphism. To a single operation of g there may correspond either a single operation of G or more than one. In the first case the isomorphism is spoken of as _simple_, in the second as _multiple_. Two simply isomorphic groups considered abstractly--that is to say, in regard only to the way in which their operations combine among themselves, and apart from any concrete representation of the operations--are clearly indistinguishable. If G is multiply isomorphic with g, let A, B, C, ... be the operations of G which correspond to the identical operation of g. Then to the operations A^(-1) and AB of G there corresponds the identical operation of g; so that A, B, C, ... constitute a subgroup H of G. Moreover, if R is any operation of G, the identical operation of g corresponds to every operation of R^(-1)HR, and therefore H is a self-conjugate subgroup of G. Since S corresponds to s, and every operation of H to the identical operation of g, therefore every operation of the set SA, SB, SC, ..., which is represented by SH, corresponds to s. Also these are the only operations that correspond to s. The operations of G may therefore be divided into sets, no two of which contain a common operation, such that the correspondence between the operations of G and g connects each of the sets H, SH, TH, UH, ... with the single operations 1, s, t, u, ... written below them. The sets into which the operations of G are thus divided combine among themselves by exactly the same laws as the operations of g. For if st = u, then SH.TH = UH, in the sense that any operation of the set SH followed by any operation of the set TH gives an operation of the set UH. The group g, abstractly considered, is therefore completely defined by the division of the operations of G into sets in respect of the self-conjugate subgroup H. From this point of view it is spoken of as the _factor-group_ of G in respect of H, and is represented by the symbol G/H. Any composite group in a similar way defines abstractly a factor-group in respect of each of its self-conjugate subgroups.