and a group of type ln it cannot have the property that each of its subgroups of composite order is invariant. If the order of G is not a power of a prime number each of its Sylow subgroups whose order exceeds a prime number is invariant under G and hence G involves invariantly the direct product of all its Sylow subgroups of composite orders. The quotient group of G with respect to this direct...

Let p be a prime,G a finite group which has a normal p-subgroup containing its own centralizer inG, and R a commutative local ring with residue class field of characteristic p. In this paper, it is shown that if α is an augmented automorphism of RG which fixes a Sylow p-subgroup P of G, there is ρ ∈ Aut(G) such that xαρ = x for all x ∈ P , and αρ is an inner automorphism of RG.

There is a longstanding conjecture, due to Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. The most successful approach to this conjecture has been Borovik’s program analyzing a minimal counterexample, or simple K-group. We show that a simple K-group of finite Morley rank and odd type is either aalgebraic of else has Prufer rank at mos...

We show that a tower of torsors under affine group schemes can be dominated by a torsor. Moreover, if the base is the spectrum of a field and the structure group schemes are finite, the tower can be dominated by a finite torsor. As an application, we show that if X is a torsor under a finite group scheme G over a scheme S which has a fundamental group scheme, then X has a fundamental group sche...

We prove that a finite group G $G$ has normal Sylow p $p$ -subgroup P $P$ if, and only every irreducible character of appearing in the permutation ( 1 ) $({\bf 1}_P)^G$ with multiplicity coprime to degree . This confirms prediction by Malle Navarro from 2012. Our proof above result depends on reduction simple groups ultimately combinatorial analysis properties branching coefficients for symmetr...

Let π1(C) be the algebraic fundamental group of a smooth connected affine curve, defined over an algebraically closed field of characteristic p > 0 of countable cardinality. Let N be a normal (resp. characteristic) subgroup of π1(C). Under the hypothesis that the quotient π1(C)/N admits an infinitely generated Sylow p-subgroup, we prove that N is indeed isomorphic to a normal (resp. characteris...

It is known that the Sylow subgroups of a Frobenius complement are cyclic or generalized quaternion. In this paper it is shown that there are no restrictions at all on the structure of the Sylow subgroups of the FrobeniusWielandt complements that appear in the well-known Wielandt's generalization of Frobenius' Theorem. Some examples of explicit constructions are also given. 0. Introduction Let ...