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Finite groups with graphs containing no triangles. (English) Zbl 1024.20021

For a given finite group \(G\) define a graph \(\Gamma(G)\) by taking the non-central conjugacy classes of \(G\) as vertices and connecting any two vertices \(A\) and \(B\) whenever \(|A|\) and \(|B|\) have a common prime divisor.
By studying the properties of \(\Gamma(G)\) it is expected to gain some useful information about \(G\) and vice versa. The purpose of the paper is to prove the following Theorem. Let \(G\) be a non-Abelian finite group. Then \(\Gamma(G)\) is a graph without triangles if and only if \(G\) is isomorphic to one of the following solvable groups: (1) the symmetric group \(S_3\); (2) a dihedral group of order \(10\) or \(12\); (3) the alternating group \(A_4\); (4) the group \(T_{12}:=\langle a,b\mid a^6=1,\;b^2=a^3,\;ba=a^{-1}b\rangle\) of order \(12\); (5) the group \(T_{21}:=\langle a,b\mid a^3=b^7=1,\;ba=ab^2\rangle\) of order \(21\). Furthermore, the property that \(\Gamma(G)\) has no triangles is equivalent to the one that \(\Gamma(G)\) is a disjoint union of two connected trees.

MSC:

20D60 Arithmetic and combinatorial problems involving abstract finite groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20E45 Conjugacy classes for groups
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