The probability of generating a finite simple group

摘要

We study the probability of generating a finite simple group, together with its generalisation P G,socG (d), the conditional probability of generating an almost simple finite group G by d elements, given that these elements generate G/socG. We prove that P G,socG (2) ⩽ 53/90, with equality if and only if G is A6 or S6, and establish a similar result for P G,socG (3). Positive answers to longstanding questions of Wiegold on direct products, and of Mel’nikov on profinite groups, follow easily from our results.