The diameter of the generating graph of a finite soluble group

[摘要] Let G be a finite 2-generated soluble group and suppose that < a(1), b(1)>= < a(2), b(2)> = G. Then there exist c(1), c(2) such that < a(1), b(1)> = < c(1), c(1)> = < c(2), c(2)> = G. Equivalently, the subgraph Delta(G) of the generating graph of a 2-generated finite soluble group G obtained by removing the isolated vertices has diameter at most 3. We construct a-generated group G of order 2(10).3(2) for which this bound is sharp. However a stronger result holds if G' has odd order or G' is nilpotent: in this case there exists b is an element of G with < a(1), b > = < a(1), b > = G. (C) 2017 Elsevier Inc. All rights reserved.

[发布日期] 2017-12-15

[关键词] Soluble groups;Generating graph