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Finite equilibrated groups. (English) Zbl 0891.20015
The titular \(E\)-groups are finite groups \(G\) in which no subgroup \(G_1\) can be expressed as the product of non-normal subgroups of \(G_1\). The authors determine the simple nonabelian \(E\)-groups: the \(L_2(p)\) where \(p\) is a prime, \(p\equiv 5 \bmod 8\), \(p^2\equiv-1\bmod 5\), \(p+1\) is twice a prime power and \(p-1\) is four times a prime power. And solvable \(E\)-groups \(C\) have many nice properties: \(G\) is an extension of a \(p\)-group by a Dedekind group and the order of \(G'\) is either a prime power or twice a prime power. In Section 4 \(E\)-groups of prime power order are investigated and it is shown that for \(p>3\) all except one are of class at most 2. It is noted that the situation looks more complicated for \(p=2\).

20D40 Products of subgroups of abstract finite groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D15 Finite nilpotent groups, \(p\)-groups
20D05 Finite simple groups and their classification
Full Text: DOI
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