Finite groups with graphs containing no triangles
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.
scientific article; zbMATH DE number 1956753