Groups with Operators
In this chapter we resume our study of the theory of groups. The results that we obtain concern the correspondence between the subgroups of a group and those of a homomorphic image, normal series and composition series, the Schreier theorem, direct products and the Krull-Schmidt theorem. The range of application of these results is enormously extended by introducing the new concept of a group with operators. This concept, which was first considered by Krull and by Emmy Noether, enables one to study a group relative to an arbitrary set of endomorphisms. In this way, one achieves a uniform derivation of a number of classical results that were formerly derived separately. Also applications to the theory of rings are obtained by considering the additive group relative to the sets of multiplications as operator domains.
KeywordsDirect Product Factor Group Homomorphic Image Chain Condition Subdirect Product
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