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Now OSS′ has a meaning only if OS is an object on which S′ may be performed. Then OST is a definite operation of the set, and therefore the result of carrying out S and then T on the set of objects is some operation U with a unique result.Hence whatever object of the set O may be, both OS and OS′ belong to the set. Represent by U′ the result of carrying out T′ and then S′.A group is spoken of as simple when it has no self-conjugate subgroup other than that constituted by the identical operation alone.

The conception of operations being conjugate to each other is extended to subgroups. are the operations of a subgroup H, and if R is any operation of G, then the operations RU′R. In particular, it is necessarily the same as H if R belongs to H.

If it is not identical with H, it is said to be conjugate to H; and it is in any case represented by the symbol RHR, when for R is taken in turn each of the operations of G, then H is called a self-conjugate subgroup of G.

For example, the operations of squaring and extracting the square root are definite inverse operations if the objects are restricted to be real positive numbers, but not otherwise. is the totality of the objects on which a definite operation S and its inverse S′ may be carried out, and if the result of carrying out S on O is represented by OS, then OSS′, OS′S, and O are the same object whatever object of the set O may be.

This will be represented by the equations SS′ = S′S = 1. Suppose now that T is another definite operation with the same set of objects as S, and that T′ is its inverse operation.

Thus in elementary arithmetic there are the fundamental operations of the addition and the multiplication of integers; in algebra a linear transformation is an operation which may be carried out on any set of variables; while in geometry a translation, a rotation, or a projective transformation are operations which may be carried out on any figure.

In speaking of an operation, an object or a set of objects to which it may be applied is postulated; and the operation may, and generally will, have no meaning except in regard to such a set of objects.

If S′ is the inverse operation of S, a group which contains S must contain SS′, which produces no change on any possible object. can be chosen which themselves constitute a group H, the group H is called a subgroup of G.

This is called the identical operation, and will always be represented by I. Thus, in particular, if S is an operation of G, the cyclical group constituted by ..., SST or TS = ST, S and T are called permutable operations.

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.

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