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

Freedman, Saul D.

Algebraic Combinatorics, Tome 6 (2023) no. 5, pp. 1395-1418

Let $G$ be a (finite or infinite) group such that $G/Z\left(G\right)$ is not simple. The non-commuting, non-generating graph $\Xi \left(G\right)$ of $G$ has vertex set $G\setminus Z\left(G\right)$, with vertices $x$ and $y$ adjacent whenever $[x,y]\ne 1$ and $\langle x,y\rangle \ne G$. We investigate the relationship between the structure of $G$ and the connectedness and diameter of $\Xi \left(G\right)$. 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 $G/Z\left(G\right)$ is finite and simple.

Classification : 20D60, 20F16, 05C25
Keywords: non-commuting non-generating graph, soluble groups, generating graph, graphs defined on groups

[1] Aalipour, Ghodratollah; Akbari, Saieed; Cameron, Peter J.; Nikandish, Reza; Shaveisi, Farzad On the structure of the power graph and the enhanced power graph of a group, Electron. J. Combin., Volume 24 (2017) no. 3, 3.16, 18 pages | DOI | MR | Zbl

[2] Abdollahi, A.; Akbari, S.; Maimani, H. R. Non-commuting graph of a group, J. Algebra, Volume 298 (2006) no. 2, pp. 468-492 | DOI | MR | Zbl

[3] Adnan, Saad On groups having exactly $2$ conjugacy classes of maximal subgroups. II, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), Volume 68 (1980) no. 3, p. 179 | MR | Zbl

[4] Belk, James Michael Thompson’s group $F$, Ph. D. Thesis, Cornell University, August (2004)

[5] Besche, Hans Ulrich; Eick, Bettina; O’Brien, E. A. The groups of order at most 2000, Electron. Res. Announc. Amer. Math. Soc., Volume 7 (2001), pp. 1-4 | DOI | MR | Zbl

[6] Biswas, Sucharita; Cameron, Peter J.; Das, Angsuman; Dey, Hiranya Kishore On the difference of the enhanced power graph and the power graph of a finite group (2022) | arXiv

[7] Bosma, Wieb; Cannon, John; Playoust, Catherine The Magma algebra system. I. The user language, J. Symbolic Comput., Volume 24 (1997) no. 3-4, pp. 235-265 | DOI | MR | Zbl

[8] Breuer, Thomas; Guralnick, Robert M.; Kantor, William M. Probabilistic generation of finite simple groups. II, J. Algebra, Volume 320 (2008) no. 2, pp. 443-494 | DOI | MR | Zbl

[9] Burness, Timothy; Guralnick, Robert; Harper, Scott The spread of a finite group, Ann. of Math. (2), Volume 193 (2021) no. 2, pp. 619-687 | DOI | MR | Zbl

[10] Cameron, Peter J. Graphs defined on groups, Int. J. Group Theory, Volume 11 (2022) no. 2, pp. 53-107 | DOI | MR | Zbl

[11] Cameron, Peter J.; Freedman, Saul D.; Roney-Dougal, Colva M. The non-commuting, non-generating graph of a nilpotent group, Electron. J. Combin., Volume 28 (2021) no. 1, 1.16, 15 pages | DOI | MR | Zbl

[12] Cohn, P. M. Basic algebra: groups, rings and fields, Springer-Verlag London, Ltd., London, 2003, xii+465 pages | DOI | MR

[13] Cox, Charles Garnet A note on the $R_{\infty }$ property for groups $\mathrm{FAlt}\left(X\right)⩽G⩽\mathrm{Sym}\left(X\right)$, Comm. Algebra, Volume 47 (2019) no. 3, pp. 978-989 | DOI | MR | Zbl

[14] Cox, Charles Garnet On the spread of infinite groups, Proc. Edinb. Math. Soc. (2), Volume 65 (2022) no. 1, pp. 214-228 | DOI | MR | Zbl

[15] Crestani, Eleonora; Lucchini, Andrea The non-isolated vertices in the generating graph of a direct powers of simple groups, J. Algebraic Combin., Volume 37 (2013) no. 2, pp. 249-263 | DOI | MR | Zbl

[16] Freedman, Saul D. Diameters of graphs related to groups and base sizes of primitive groups, Ph. D. Thesis, University of St Andrews, May (2022)

[17] Freedman, Saul D. The non-commuting, non-generating graph of a finite simple group (2022 (submitted)) | arXiv

[18] Hering, Christoph Transitive linear groups and linear groups which contain irreducible subgroups of prime order, Geometriae Dedicata, Volume 2 (1974), pp. 425-460 | DOI | MR | Zbl

[19] Herstein, I. N. A remark on finite groups, Proc. Amer. Math. Soc., Volume 9 (1958), pp. 255-257 | DOI | MR | Zbl

[20] Herzog, Marcel; Longobardi, Patrizia; Maj, Mercede On a graph related to the maximal subgroups of a group, Bull. Aust. Math. Soc., Volume 81 (2010) no. 2, pp. 317-328 | DOI | MR | Zbl

[21] Liebeck, Martin W.; Shalev, Aner Simple groups, probabilistic methods, and a conjecture of Kantor and Lubotzky, J. Algebra, Volume 184 (1996) no. 1, pp. 31-57 | DOI | MR | Zbl

[22] Lucchini, Andrea The diameter of the generating graph of a finite soluble group, J. Algebra, Volume 492 (2017), pp. 28-43 | DOI | MR | Zbl

[23] Lucchini, Andrea; Maróti, Attila; Roney-Dougal, Colva M. On the generating graph of a simple group, J. Aust. Math. Soc., Volume 103 (2017) no. 1, pp. 91-103 | DOI | MR | Zbl

[24] Ore, Oystein Contributions to the theory of groups of finite order, Duke Math. J., Volume 5 (1939) no. 2, pp. 431-460 | DOI | MR | Zbl

[25] Shi, Wu Jie Finite groups having at most two classes of maximal subgroups of the same order, Chinese Ann. Math. Ser. A, Volume 10 (1989) no. 5, pp. 532-537 | MR

[26] Smith, Simon M. A classification of primitive permutation groups with finite stabilizers, J. Algebra, Volume 432 (2015), pp. 12-21 | DOI | MR | Zbl

[27] Suzuki, Michio On a class of doubly transitive groups, Ann. of Math. (2), Volume 75 (1962), pp. 105-145 | DOI | MR | Zbl