The non-commuting, non-generating graph of a non-simple group

Saul Daniel Freedman*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Let G be a (finite or infinite) group such that G/Z(G) is not simple. The non- commuting, non-generating graph Ξ(G) of G has vertex set G\Z(G), with vertices x and y adjacent whenever [x, y]̸ ≠ 1 and ⟨x, y⟩̸ ≠ G. We investigate the relationship between the structure of G and the connectedness and diameter of Ξ(G). In particular, we prove that the graph either: (i) is connected with diameter at most 4; (ii) consists of isolated vertices and a connected component of diameter at most 4; or (iii) is the union of two connected components of diameter 2. We also describe in detail the finite groups with graphs of type (iii). In the companion paper [17], we consider the case where G/Z(G) is finite and simple.
Original languageEnglish
Pages (from-to)1395-1418
Number of pages24
JournalAlgebraic Combinatorics
Issue number5
Publication statusPublished - 7 Nov 2023


  • Non-commuting non-generating graph
  • Soluble groups
  • Generating graph
  • Graphs defined on groups


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