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A local approach to certain classes of finite groups. (English) Zbl 1041.20013

A group is called a ‘PST-group’, ‘PT-group’ or ‘T-group’ if Sylow-permutability, permutability or normality, respectively, is a transitive relation. (A subgroup is ‘Sylow-permutable’, or ‘S-permutable’, in a group if it permutes with every Sylow subgroup.)
The classes of finite PST-groups, PT-groups and T-groups have been widely studied in recent years. For example, structural characterizations have been given by the reviewer [in J. Aust. Math. Soc. 70, No. 2, 143-159 (2001; Zbl 0997.20027)].
In the present article the authors provide local characterizations of the three classes, i.e., by properties that refer to individual primes. (All groups considered are finite.)
Let \(G\) be a group and \(p\) a prime. Then \(G\) satisfies the condition \(N_p\) if, for all \(N\triangleleft G\), subgroups of \(O_p(G/N)\) are S-permutable in \(G/N\) and non-Abelian chief factors of \(G/N\) with order divisible by \(p\) are simple. The properties \(M_p\) and \(L_p\) are defined similarly, by replacing S-permutable by permutable or normal, respectively. The main result is the following local characterization.
Theorem A. A group \(G\) is a PST-, PT- or T-group if and only if it has \(N_p\), \(M_p\), \(L_p\), respectively, for all primes \(p\).
Three further local properties \(T_p''\), \(T_p'\), \(T_p\) are studied. Here a group \(G\) has \(T_p''\) if every \(p\)-perfect subnormal subgroup is S-permutable, with corresponding definitions of \(T_p'\) and \(T_p\).
Theorem B. A group is a PST-, PT- or T-group if and only if it satisfies \(T_p''\), \(T_p'\), \(T_p\), respectively, for all primes \(p\). The relation between the two sets of local properties is also studied.

MSC:

20D35 Subnormal subgroups of abstract finite groups
20D40 Products of subgroups of abstract finite groups

Citations:

Zbl 0997.20027
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References:

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