Further Group Theory

P. M. Cohn
Additional contact information
P. M. Cohn: University College London, Department of Mathematics

Chapter 3 in Further Algebra and Applications, 2003, pp 91-134 from Springer

Abstract: Abstract Group theory has developed so much in recent years that a separate volume would be needed even for an introduction to all the main areas of research. The most a chapter can do is to give the reader a taste by a selection of topics; our choice was made on the basis of general importance or interest, and relevance in later applications. Thus ideas from extension theory (Section 3.1) are used in the study of simple algebras, while the notion of transfer (Section 3.3) has its counterpart in rings in the form of determinants. Hall subgroups (Section 3.2) are basic in the deeper study of finite groups, the ideas of universal algebra are exemplified by free groups (Section 3.4) and linear groups (Section 3.5) lead to an important class of simple groups, as do symplectic groups (Section 3.6) and orthogonal groups (Section 3.7). We recall some standard notations from BA. If a group G is generated by a set X, we write G = gp{X}, and we put gp{X|R} for a group with generating set X and set of defining relations R For subsets X, Y of G, XY denotes the set of all products xy, where x ∈ X, y ∈ Y. We write N ⊲ G to indicate that N is a normal subgroup in G, i.e. mapped into itself by all inner automorphisms of G. If H, K are subgroups of G, then HK is a subgroup precisely when HK = KH; in particular this holds when H or K is normal in G. We also recall the modular law: given subgroups K, L, M of G, if K ⊂ M, then K (L ∩ M) = KL ∩ M.

Keywords: Normal Subgroup; Span Tree; Finite Group; Minimal Normal Subgroup; Symplectic Group (search for similar items in EconPapers)
Date: 2003
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4471-0039-3_3

Ordering information: This item can be ordered from
http://www.springer.com/9781447100393

DOI: 10.1007/978-1-4471-0039-3_3

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().