Graphs and Groups, Geometries and GAP (G2G2) Summer School - External Satellite Conference of 8ECM

27 June 2021 to 3 July 2021
Rogla, Slovenia
UTC timezone

On finite non-solvable 4-primary groups without elements of order 6

Not scheduled
20m
Rogla, Slovenia

Rogla, Slovenia

Hotel Planja, Rogla 1, 3214 Zreče, Slovenia
Oral presentation

Speaker

Mr Nikolai Minigulov (Krasovskii Institute of Mathematics and Mechanics UB RAS)

Description

We use mainly standard notation and terminology (see [1]).

Let $G$ be a finite group. Denote by $\pi(G)$ the set of all prime divisors of the order of $G$. If $|\pi(G)| = n$, then $G$ is called $n$-primary.

The Gruenberg--Kegel graph (prime graph) $\Gamma(G)$ of $G$ is a graph with the vertex set $\pi(G)$, in which two distinct vertices $p$ and $q$ are adjacent if and only if there exists an element of order $pq$ in $G$.

In 2012–2013, A.S. Kondrat'ev described finite groups having the same Gruenberg-Kegel graph as the groups $Aut(J_2)$ [2] and $A_{10}$ [3], respectively. The Gruenberg--Kegel graphs of these groups are isomorphic as abstract graphs.

We establish a more general problem: to describe finite groups whose Gruenberg-Kegel graphs are isomorphic as abstract graphs to the graph $\Gamma(A_{10})$.

As a part of the solution of this problem, we proved in [4] that if $G$ is a finite non-solvable group and the graph $\Gamma(G)$ as abstract graph is isomorphic to the graph $\Gamma(A_{10})$, then the quotient group $G/S(G)$ (where $S(G)$ is the solvable radical of $G$) is almost simple, and classified all finite almost simple groups whose the Gruenberg-Kegel graphs as abstract graphs are isomorphic to subgraphs of $\Gamma(A_{10})$.

Let $G$ be a finite non-solvable group, and the graph $\Gamma(G)$ as abstract graph is isomorphic to the graph $\Gamma(A_{10})$. Then the graph $\Gamma(G)$ has the following form (see attachment pdf), where $p$, $q$, $r$, and $s$ are pairwise distinct primes.

In [5], we described such finite non-solvable groups $G$ when 3 does not divide $|G|$.

In this work, we prove the following theorem.

Theorem. Let $G$ be a finite non-solvable group, $\overline{G}\cong G/S(G)$ and $\Gamma(G)$ as abstract graph is isomorphic to $\Gamma{(A_{10})}$. If $3$ divides of $|G|$ and $G$ has no elements of order $6$, then one of the following statements holds:

$(1)$ $S=O_{2',2}(G)$, $O(G)=O_p(G)$, $q=2$, $S/O(G)$ is an elementary abelian $2$-group, $\overline{G}\cong{L_2(2^n)}$ and one of the following statements holds:

$(1a)$ $n=4$, $p=17$ and $\{r,s\}=\{3,5\}$;

$(1b)$ $n$ is prime, $n\ge5$, $p=2^n-1$, and $\{r,s\}=\{3,(2^n-1)/3\}$, the group $S/O(G)$ either is trivial or as $\overline{G}$-module is isomorphic to a direct sum of natural $GF(2^n)\overline{G}$-modules;

$(2)$ $S=O_p(G)$, $q=2$, $\overline{G}\cong L_2(p)$, $p\ge31$, $p\equiv\varepsilon5(mod$ $12)$, $\varepsilon\in\{+,-\}$, $p-\varepsilon1=2^k$, and $3\in\{r,s\}=\pi((t+\varepsilon1)/2)$;

$(3)$ $S=O_p(G)$, $q=3$, and one of the following statements holds:

$(3a)$ $\overline{G}\cong PGL_2(9)$, $p>5$, and $\{r,s\}=\{2,5\}$;

$(3b)$ $\overline{G}\cong L_2(81)$, $PGL_2(81)$ or $L_2(81).2_3$, $p=41$, and $\{r,s\}=\{2,5\}$;

$(3c)$ $\overline{G}\cong L_2(3^n)$ or $PGL_2(3^n)$, $n$ is an odd prime, $p=(3^n-1)/2$, and $\{r,s\}=\pi(3^n+1)$.

Each of the statements $(1)$--$(3)$ of the theorem is realized.

In the proof of Theorem, we use the classification of the finite non-solvable groups without elements of order $6$ from (Theorem 2, [6]).

Acknowledgments. The work is supported by Russian Science Foundation (project 19-71-10067).

References:
[1] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of finite groups, Clarendon Press, Oxford (1985).
[2] A. S. Kondrat'ev, Finite groups with prime graph as in the group $Aut(J_2)$. Proc. Steklov Inst. Math. 283(1) (2013), 78--85.
[3] A. S. Kondrat'ev, Finite groups that have the same prime graph as the group $A_{10}$. Proc. Steklov Inst. Math. 285(1) (2014), 99--107.
[4] A. S. Kondrat'ev, N. A. Minigulov, Finite almost simple groups whose Gruenberg--Kegel graphs as abstract graphs are isomorphic to subgraphs of the Gruenberg--Kegel graph of the alternating group $A_{10}$. Siberian Electr. Math. Rep. 15 (2018), 1378--1382.
[5] A. S. Kondrat'ev, N. A. Minigulov, On finite non-solvable $4$-primary $3'$-groups, in: Algebra, number theory and mathematical modeling of dynamical systems: abstracts of the international conference devoted to the $70$th anniversary of A.Kh. Zhurtov, KBSU, Nal'chik, 2019, 56 (in Russian).
[6] A. S. Kondrat'ev, N. A. Minigulov, Finite groups without elements of order six. Math. Notes 104:5 (2018), 696--701.

Primary authors

Prof. Anatoly Kondrat'ev (Krasovskii Institute of Mathematics and Mechanics UB RAS) Mr Nikolai Minigulov (Krasovskii Institute of Mathematics and Mechanics UB RAS)