In recent work, Cameron, Manna and Mehatari have studied the finite groups whose power graph is a cograph, which we refer to as power-cograph groups. They classify the nilpotent groups with this property, and they establish partial results in the general setting, highlighting certain number-theoretic difficulties that arise for the simple groups of the form PSL 2 ( q ) \operatorname{PSL}_{2}(q) or Sz ( 2 2 e + 1 ) \operatorname{Sz}(2^{2e+1}) . In this paper, we prove that these number-theoretic problems are in fact the only obstacles to the classification of non-solvable power-cograph groups. Specifically, for the non-solvable case, we give a classification of power-cograph groups in terms of such groups isomorphic to PSL 2 ( q ) \operatorname{PSL}_{2}(q) or Sz ( 2 2 e + 1 ) \operatorname{Sz}(2^{2e+1}) . For the solvable case, we are able to precisely describe the structure of solvable power-cograph groups. We obtain a complete classification of solvable power-cograph groups whose Gruenberg–Kegel graph is connected. Moreover, we reduce the case where the Gruenberg–Kegel graph is disconnected to the classification of 𝑝-groups admitting fixed-point-free automorphisms of prime power order, which is in general an open problem.
We construct the first infinite families of locally 2-arc transitive graphs with the property that the automorphism group has two orbits on vertices and is quasiprimitive on exactly one orbit, of twisted wreath type. This work contributes to Giudici, Li and Praeger’s program for the classification of locally 2-arc transitive graphs by showing that the star normal quotient twisted wreath category also contains infinitely many graphs.
We construct the first infinite families of locally arc transitive graphs with the property that the automorphism group has two orbits on vertices and is quasiprimitive on exactly one orbit, of twisted wreath type. This work contributes to Giudici, Li and Praeger's program for the classification of locally arc transitive graphs by showing that the star normal quotient twisted wreath category also contains infinitely many graphs.
Cameron, Manna and Mehatari investigated the question of which finite groups admit a power graph that is a cograph, also called power-cograph groups (Journal of Algebra 591 ( 2022)). They give a classification for nilpotent groups and partial results for general groups. However, the authors point out number theoretic obstacles towards a classification. These arise when the groups are assumed to be isomorphic to PSL 2 (q) or Sz(q) and are likely to be hard.In this paper, we prove that these number theoretic problems are in fact the only obstacles to the classification of non-solvable power-cograph groups. Specifically, for the non-solvable case, we give a classification of power-cograph groups in terms of such groups isomorphic to PSL 2 (q) or Sz(q). For the solvable case, we are able to precisely describe the structure of solvable power-cograph groups. We obtain a complete classification of solvable power-cograph groups whose Gruenberg-Kegel graph is connected. Moreover, we reduce the case where the Gruenberg-Kegel graph is disconnected to the classification of p-groups admitting fixed-point-free automorphisms of prime power order, which is in general an open problem. * The research leading to these results has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (EngageS: grant agreement No. 820148) and from the German Research Foundation DFG (SFB-TRR 195 "Symbolic Tools in Mathematics and their Application"). We thank Pascal Schweitzer for the helpful comments on an earlier draft of this paper.
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