Linear and Nonlinear Group Actions , and The

One of the simplest and most useful notions in mathematics is that of a group action: if G is a group and X is a nonempty set, then an action of G o n X or a G-set structure on X consists of a multiplication operation G X ! X, with the image of a pair g;x written as, say, gx, with the following axioms satissed: 1 1x = x for all x 2 X here 1 2 G is the identity element o f G; 2 ghx = ghx for all g;h2 G and all x 2 X. Equivalently, a group action determines, and is determined by, a homo-morphism g 7 ! x 7 ! gx o f G into the group of all bijective maps of X onto itself thèsymmetric' group of all permutations of X. If X has the additional structure of a linear space over a eld, we might impose an additional axiom that the mappings x 7 ,! gx all be linear. In this case, we say the action of G is linear; or that it deenes a linear representation of G, and we call X a module for G over k. In the linear or nonlinear case, if G and X have some topological structure, we often require that the map GX ,! X be continuous, or in the algebraic geometry case, a morphism of algebraic varieties or schemes. A main theme of the Newton Institute program 1 on the representation theory of algebraic and related nite groups, of which this conference volume is a part, may then be stated as follows: even in the simplest case where G is a nite group, and X is a nite set with no linear space or algebraic variety structure, there are, nevertheless, deep connections of these nonlinear actions with linear actions, and of the discrete theory of nite groups and their actions with the theory of algebraic groups and other groups and algebras arising in continuous Lie theory. I hope to explain these connections 1 The author would like to thank the NSF for its support and the Newton Institute for its hospitality.