Centralisers of Involutions in Black Box Groups

We discuss basic structural properties of finite black box groups. A special emphasis is made on the use of centralisers of involutions in probabilistic recognition of black box groups. In particular, we suggest an algorithm for finding the p-core of a black box group of odd characteristic. This special role of involutions suggest that the theory of black box groups reproduces, at a non-deterministic level, some important features of the classification of finite simple groups. 1. What is a black box group? A black box group X is a device or an algorithm (‘oracle’ or ‘black box’ ) which produces (nearly) uniformly distributed independent random elements from some finite group X. These elements are encoded as 0–1 strings of uniform length; given strings representing x, y ∈ X, the black box can compute strings representing xy and x−1, and decide whether x = y in time bounded from above by a constant. In this setting, one is usually interested in finding probabilistic algorithms which allow us to determine, with probability of error , the isomorphism type of X in time O(| | · (log|X|)). We say in this situation that our algorithm is run in Monte Carlo polynomial time. A critical discussion of this concept can be found in [6], while [7] contains a detailed survey of the subject. See also the forthcoming book by Seress [38]. In this paper we discuss a (still rather rudimentary) structural approach to the theory of black box group. We briefly survey methods for constructing black box oracles for subgroups and factor groups of black box groups, and then show how one can construct black box oracles for centralisers of involutions. They are used in the algorithm for finding the p-core of a black box group of characteristic p. Isomorphisms and homomorphisms of black box groups are understood as isomorphisms and homomorphisms of their underlying groups. However we reserve the term black box subgroup for a subgroup of a black box group endowed with its own black box oracle. Despite this rather abstract general setting, practically important black box groups usually appear as big permutation or matrix groups. For example, given 1991 Mathematics Subject Classification. Primary 51F15; Secondary 20G40.