Further Group Theory

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.