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Let be a finite group and write for the degree set of the complex irreducible characters of . The group is said to satisfy the two-prime hypothesis if for any distinct degrees , the total number of (not necessarily different) primes of the greatest common divisor is at most . We prove an upper bound on the number of irreducible character degrees of a nonsolvable group that has a composition factor isomorphic to PSL for .
A family of loops is studied, which arises with its binary operation in a natural way from some transversals possessing a ``normality condition''.
This paper deals with a rationality condition for groups. Let n be a fixed positive integer. Suppose every element g of the finite solvable group is conjugate to its nth power g n. Let p be a prime divisor of the order of the group. We conclude that the multiplicative order of n modulo p is small, or p is small.
This paper studies groups G whose all subgroups are either ascendant or self-normalizing. We characterize the structure of such G in case they are locally finite. If G is a hyperabelian group and has the property, we show that every subgroup of G is in fact ascendant provided G is locally nilpotent or non-periodic. We also restrict our study replacing ascendant subgroups by permutable subgroups, which of course are ascendant [Stonehewer S.E., Permutable subgroups of infinite groups, Math. Z., 1972,...
Our main result is that a locally graded group whose proper subgroups are Baer-by-Chernikov is itself Baer-by-Chernikov. We prove also that a locally (soluble-by-finite) group whose proper subgroups are Baer-by-(finite rank) is itself Baer-by-(finite rank) if either it is locally of finite rank but not locally finite or it has no infinite simple images.
If is a class of groups, then a group is said to be minimal non -group if all its proper subgroups are in the class , but itself is not an -group. The main result of this note is that if is an integer and if is a minimal non (respectively, )-group, then is a finitely generated perfect group which has no non-trivial finite factor and such that is an infinite simple group; where (respectively, , ) denotes the class of nilpotent (respectively, nilpotent of class at most , locally...
In this paper we investigate the structure of X-groups in which every subgroup is permutable or of finite rank. We show that every subgroup of such a group is permutable.
A subgroup H of a group G is said to be nearly normal in G if it has finite index in its normal closure in G. A well-known theorem of B.H. Neumann states that every subgroup of a group G is nearly normal if and only if the commutator subgroup G' is finite. In this article, groups in which the intersection and the join of each system of nearly normal subgroups are likewise nearly normal are considered, and some sufficient conditions for such groups to be finite-by-abelian are given.
A subgroup H of a group G is said to be quasinormal if HX =
XH for all subgroups X of G. In this article groups are characterized for
which the partially ordered set of quasinormal subgroups is decomposable.
A subgroup H of a group G is called ascendant-by-finite in G if there exists a subgroup K of H such that K is ascendant in G and the index of K in H is finite. It is proved that a locally finite group with every subgroup ascendant-by-finite is locally nilpotent-by-finite. As a consequence, it is shown that the Gruenberg radical has finite index in the whole group.
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