Insoluble groups with the rewriting property P8

Journal of Algebra, 1988

Illinois Journal of Mathematics - ILL J MATH, 1970

Journal of the London Mathematical Society, 1990

Journal of Algebra, 1978

Mathematische Zeitschrift, 1981

Journal of Algebra, 2006

Let M be a maximal subgroup of a finite group G and K/L be a chief factor such that L ≤ M while K M. We call the group M ∩ K/L a c-section of M. And we define Sec(M) to be the abstract group that is isomorphic to a c-section of M. For every maximal subgroup M of G, assume that Sec(M) is supersolvable. Then any composition factor of G is isomorphic to L 2 (p) or Z q , where p and q are primes, and p ≡ ±1(mod 8). This result answer a question posed by ref. [12].

Journal of Algebra, 1980

A well-known theorem of Wielandt states that a finite group G is nilpotent if and only if every maximal subgroup of G is normal in G. The structure of a nonnilpotent group, each of whose proper subgroups is nilpotent, has been analyzed by Schmidt and R6dei [5, Satz 5.1 and Satz 5.2, pp. 280-281]. In [1], Buckley investigated the structure of a PN-group (i.e., a finite group in which every minimal subgroup is normal), and proved (i) that a PN-group of odd order is supersolvable, and (ii) that certain factor groups of a PN-group of odd prime power order are also PN-groups. Earlier, Gaschiitz and It5 [5, Satz 5.7, p. 436] had proved that the commutator subgroup of a finite PN-group is p-nilpotent for each odd prime p. This paper is a sequel to [9] and our object here is to prove the following statement. THEOREM. If G is a finite nonPN-group, each of whose proper subgroups is a PN-group, then one of the following statements is true: (a) G is the dihedral group of order 8.

2003

We give an example of an infinite simple Frobenius group G without involutions, with a trivial kernel and a nilpotent complement. Nevertheless, this group is not $\omega $- stable (not even superstable), this is the "only" property missing in order to be a counterexample to the Cherlin-Zil'ber Conjecture which says that simple $\omega $- stable groups are algebraic groups.

Pre-publicaciones del …, 2004

We study the relationship between Carter subgroups and groups satisfying the minimal condition on their abnormal subgroups in some classes of generalized soluble groups.

Let M be a maximal subgroup of a finite group G and K/L be a chief factor such that L ≤ M while K M. We call the group M ∩ K/L a c-section of M. And we define Sec(M) to be the abstract group that is isomorphic to a c-section of M. For every maximal subgroup M of G, assume that Sec(M) is supersolvable. Then any composition factor of G is isomorphic to L 2 (p) or Z q , where p and q are primes, and p ≡ ±1(mod 8). This result answer a question posed by ref. [12].