Finite groups of order 2a3b17c, III (original) (raw)
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Algebra and Logic, 1980
In the recent past a series of strong results have been announced, which essentially constitute an exhaustive treatment of the problem of describing the p -local structure of finite groups p of type characteristic two in the case where the G -rank ( p an odd prime), of the 2-local subgroups of G is sufficiently big (viz°, ~ ). The situation is much less clear in the case of small fl -rank. Here, it seems that a characterization would be useful of known simple groups, not necessarily of type characteristic two, by means of the centralizers of elements of order /D, or -in the first place -by means of the centralizers of elements of order three.
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.
A note on the solvability of groups
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].
Simple groups of order p · 3a · 2b
Journal of Algebra, 1970
In this paper some local group theoretic properties of a simple group G of order p * 3@ * 2b are found. These are applied in a later paper to show there are no simple groups of order 7 * 3" * 2b other than the three well-known ones. R. Brauer [4] has shown there are only the three known simple groups LI, , A, , and O,(3) of order 5 .3" * 2b. His treatment uses modular character theory especially for the prime 5. It follows from J. Thompson's X-group paper [9] that if G is a simple group of order p + 3" * 2*, then p = 5,7, 13, or 17. In our later treatment of the case p '=-T 7 we seem to need the results of the N-group paper itself. This present paper prepares the way.