On finite simple nonabelian groups of Lie type over fields of different characteristics with the same prime graph
M. Zinovieva
Let \(G\) be a finite group, let \(\pi(G)\) be the set of all prime divisors of its order,
and let \(\omega(G)\) be the spectrum of \(G\), i.e. the set of its element orders. The set \(\omega(G)\) defines
a graph with the following adjacency relation: different vertices \(r\) and \(s\) in \(\pi(G)\) are joined by an edge if and only if
\(rs\in \omega(G)\). This graph is called the Gruenberg–Kegel graph or the prime graph of \(G\) and is denoted by \(GK(G)\).
In [1], A. V. Vasil'ev posed Problem 16.26:
Does there exist a positive integer \(k\) such that there are no \(k\) pairwise nonisomorphic finite nonabelian simple groups with the same prime graph?
Conjecture: \(k=5\).
It is easy to see that there exist four pairwise nonisomorphic finite nonabelian simple groups with the same prime graph, namely: \(J_2\), \(A_9\), \(C_3(2)\), \(D_4(2)\).
In [2], the case is investigated where the alternating group \(A_n\) for \(n\geq 5\) and a finite simple group have the same prime graph.
In the present abstract, we consider two finite simple groups of Lie type over fields of different characteristics \(p_1\) è \(p_2\). Using the information about the prime graphs of finite simple groups from [3–6]
and the orders of groups of Lie type, which can be found, for example, in [7], we obtain the following result.
Proposition 1.Let \(G_i=A_{n_i-1}(q_i)\), where \(n_i\geq 7\), \(n_i\) is odd, \(q_i\) is a power of an odd prime \(p_i\) for \(i\in\{1,2\}\), \(p_2\neq p_1\), and let \(GK(G_1)=GK(G_2)\). Then \(n_1=n_2\) and the following condition holds:
\(\ \ (*)\ \) either \(p_i\) is a primitive prime divisor of \({q_j}^3-1\), or \(n_{p_i}=(q_j-1)_{p_i}\) and \(p_i\) is a primitive prime divisor of \(q_j-1\) for \(\{i,j\}=\{1,2\}\).
Proposition 2. Let \(G_1=A_{n_1-1}(q_1)\), where \(n_1\geq 7\), \(n_1\) is odd, let \(G_2=B_{n_2-1}(q_2)\), where \(n_2\equiv0,1\pmod4\), \(n_1\geq8\), \(q_i\) is a power of an odd prime \(p_i\) for \(i\in\{1,2\}\),
\(p_i\) are primes, \(p_2\neq p_1\), and let \(GK(G_1)=GK(G_2)\). Then \(n_1=3n_2/2+1\) or \(n_1=3n_2/2+3/2\) and the following conditions hold:
\(\ \ (1)\ \) \(n_1\equiv4,5,9\pmod{12}\);
\(\ \ (2)\ \) \(p\) is a primitive prime divisor of \(q_1^6-1\);
\(\ \ (3)\ \) \(p_1\) is a primitive prime divisor of \(q^j-1\), \(j\in\{1,2,3\}\).
Acknowledgement. The work is supported by RFBR (project N 13-01-00469).
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