Abstract
The study of finite groups whose prime graphs do not contain triangles is continued. The main result of this paper is the following theorem: if G is a finite nonsolvable group whose prime graph contains no triangles and S(G) is the greatest solvable normal subgroup of G, then |π(G)| ≤ 8 and |π(S(G))| ≤ 3. A detailed description of the structure of a group G satisfying the conditions of the theorem is obtained in the case when π(S(G)) contains a number that does not divide the order of the group G/S(G). We also construct an example of a finite solvable group of Fitting length 5 whose prime graph is a 4-cycle. This completes the determination of the exact bound for the Fitting length of finite solvable groups whose prime graphs do not contain triangles.
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Original Russian Text © O.A. Alekseeva, A.S. Kondrat’ev, 2016, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2016, Vol. 22, No. 1.
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Alekseeva, O.A., Kondrat’ev, A.S. Finite groups whose prime graphs do not contain triangles. II. Proc. Steklov Inst. Math. 296 (Suppl 1), 19–30 (2017). https://doi.org/10.1134/S0081543817020031
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DOI: https://doi.org/10.1134/S0081543817020031