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Normal Structure of Groups

In: Algebra

Author

Listed:
  • Ernest Shult

    (Kansas State University, Department of Mathematics)

  • David Surowski

Abstract

The Jordan-Hölder Theorem for Artinian Groups is a simple application of the poset-theoretic Jordan-Hölder Theorem expounded in Chap. 2 . A discussion of commutator identities is exploited in defining the derived series and solvability as well as in defining the upper and lower central series and nilpotence. The Schur-Zassenhaus Theorem for finite groups ends the chapter. In the exercises, one will encounter the concept of normally-closed families of subgroups of a group G, which gives rise to several well-known characteristic subgroups, such as $${\mathbf O}_p(G)$$ O p ( G ) , the torsion subgroup, and (when G is finite) the Fitting subgroup. Some further challenges appear in the exercises.

Suggested Citation

  • Ernest Shult & David Surowski, 2015. "Normal Structure of Groups," Springer Books, in: Algebra, edition 127, chapter 0, pages 137-161, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-19734-0_5
    DOI: 10.1007/978-3-319-19734-0_5
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