Abstract
We study Problem 8.31 in [1] of the description of finite weakly factorizable groups, i.e., the groups whose every proper subgroup is complemented in a larger subgroup. By Lemma 1 of [2], the subdirect product of a completely factorizable group (such groups were studied by Ph. Hall and N. V. Baeva (Chernikova)) and a weakly factorizable group is a weakly factorizable group. In connection with a remark in [2], we observe that a dihedral 2-group is always weakly factorizable but, in general, it cannot even be obtained from the groups of prime order by repeated application of the lemma. Theorem 1, basing on the available maximal factorizations, shows that there exist exactly three finite simple nonabelian groups with complemented maximal subgroups. Theorem 2 confirms the conjecture of [2] on uniqueness of a finite simple nonabelian group with the property of weak factorizability. Earlier Theorems 1 and 2 were proven in special cases by A. G. Likharev.
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Original Russian Text Copyright © 2006 Levchuk V. M. and Likharev A. G.
The authors were supported by the Russian Foundation for Basic Research (Grants 03-01-00905 and 06-01-00824a).
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 47, No. 4, pp. 798–810, July–August, 2006.
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Levchuk, V.M., Likharev, A.G. Finite simple groups with complemented maximal subgroups. Sib Math J 47, 659–668 (2006). https://doi.org/10.1007/s11202-006-0077-7
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DOI: https://doi.org/10.1007/s11202-006-0077-7