Torsion completeness of <Emphasis Type="Italic">p</Emphasis>-primary components in modular group rings of <Emphasis Type="Italic">p</Emphasis>-reduced Abelian groups

Let G be a p-reduced Abelian group and R a commutative unital ring of prime characteristic p such that for each natural number i the subring $ R^{p^i } $ R^{p^i } has nilpotent elements. It is shown that if S(RG) is the normalized Sylow p-group in the group ring RG, then S(RG) is torsion-complete if and only if G is a bounded p-group. This strengthens our former results on this subject.     

Finite groups with <Emphasis Type="Italic">S</Emphasis>-supplemented <Emphasis Type="Italic">p</Emphasis>-subgroups

Consider a finite group G. A subgroup is called S-quasinormal whenever it permutes with all Sylow subgroups of G. Denote by B _sG the largest S-quasinormal subgroup of G lying in B. A subgroup B is called S-supplemented in G whenever there is a subgroup T with G = BT and B∩T ≤ B _sG . A subgroup L of G is called a quaternionic subgroup whenever G has a section A/B isomorphic to the order 8 quaternion group such that L ≤ A and L ∩ B = 1. This article is devoted to proving the following theorem.     

On <Emphasis Type="Italic">c</Emphasis>-normal maximal and minimal subgroups of Sylow <Emphasis Type="Italic">p</Emphasis>-subgroups of finite groups

A subgroup H of a finite group G is called c-normal in G if there exists a normal subgroup N of G such that G = HN and $H \cap N \leq H_{G} = {\rm core}_{G}(H)$. In this paper, we investigate the class of groups of which every maximal subgroup of its Sylow p-subgroup is c-normal and the class of groups of which some minimal subgroups of its Sylow p-subgroup is c-normal for some prime number p. Some interesting results are obtained and consequently, many known results related to p-nilpotent groups and p-supersolvable groups are generalized.     

Automorphisms of Chevalley groups of type <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">l</Emphasis></Subscript> over local rings with 1/2

In this paper, we prove that every automorphism of a Chevalley group of type B _l , l ≥ 2, over a commutative local ring with 1/2 is standard, i.e., it is a composition of ring, inner, and central automorphisms.     

Finite solvable groups in which the Sylow <Emphasis Type="Italic">p</Emphasis>-subgroups are either bicyclic or of order <Emphasis Type="Italic">p</Emphasis><Superscript>3</Superscript>

All groups considered in this paper will be finite. Our main result here is the following theorem. Let G be a solvable group in which the Sylow p-subgroups are either bicyclic or of order p ³ for any p ∈ π(G). Then the derived length of G is at most 6. In particular, if G is an A₄-free group, then the following statements are true: (1) G is a dispersive group; (2) if no prime q ∈ π(G) divides p ² + p + 1 for any prime p ∈ π(G), then G is Ore dispersive; (3) the derived length of G is at most 4.     

The classification of <Emphasis Type="Italic">p</Emphasis>-local finite groups over the extraspecial group of order <Emphasis Type="Italic">p</Emphasis><Superscript>3</Superscript> and exponent <Emphasis Type="Italic">p</Emphasis>

In this paper we classify the p-local finite groups over p¹⁺²₊, the extraspecial group of order p³ and exponent p for odd p. This study reduces to the classification of the saturated fusion systems over p¹⁺²₊, which will be characterized by the outer automorphism group, the number of -radical subgroups and the automorphism group of each nontrivial -radical subgroup. As part of this classification, we obtain three new exotic 7-local finite groups.Partially supported by MCYT grant BFM2001-2035.Partially supported by MCYT grant BFM2001-1825.Both authors have been supported by the EU grant nr HPRN-CT-1999-00119.in final form: 1 October 2003     

Finite <Emphasis Type="Italic">p</Emphasis>-central groups of height <Emphasis Type="Italic">k</Emphasis>

A finite group G is called p ⁱ -central of height k if every element of order p ⁱ of G is contained in the k ^th -term ζ _k (G) of the ascending central series of G. If p is odd, such a group has to be p-nilpotent (Thm. A). Finite p-central p-groups of height p − 2 can be seen as the dual analogue of finite potent p-groups, i.e., for such a finite p-group P the group P/Ω₁(P) is also p-central of height p − 2 (Thm. B). In such a group P, the index of P ^p is less than or equal to the order of the subgroup Ω₁(P) (Thm. C). If the Sylow p-subgroup P of a finite group G is p-central of height p − 1, p odd, and N _G (P) is p-nilpotent, then G is also p-nilpotent (Thm. D). Moreover, if G is a p-soluble finite group, p odd, and P ∈ Syl _p (G) is p-central of height p − 2, then N _G (P) controls p-fusion in G (Thm. E). It is well-known that the last two properties hold for Swan groups (see [11]).     

Pure <Emphasis Type="Italic">B</Emphasis>(<Emphasis Type="Italic">n</Emphasis>)-subgroups of completely decomposable groups

We show a simple way to determine purity for a B(n)-group contained in a completely decomposable group.     

On finite groups with special Sylow <Emphasis Type=Italic>p</Emphasis>-subgroups

We find the cases in which a finite p-soluble group with a special Sylow p-subgroup has p-length 1.     

Uniform boundedness of <Emphasis Type="Italic">p</Emphasis>-primary torsion of abelian schemes

Let k be a field finitely generated over ℚ and p a prime. The torsion conjecture (resp. p-primary torsion conjecture) for abelian varieties over k predicts that the k-rational torsion (resp. the p-primary k-rational torsion) of a d-dimensional abelian variety A over k should be bounded only in terms of k and d. These conjectures are only known for d=1. The p-primary case was proved by Y. Manin, in 1969; the general case was completed by L. Merel, in 1996, after a series of contributions by B. Mazur, S. Kamienny and others. Due to the fact that moduli of elliptic curves are 1-dimensional, the d=1 case of the torsion conjecture (resp. p-primary torsion conjecture) is closely related to the following. For any k-curve S and elliptic scheme E→S, the k-rational torsion (resp. the p-primary k-rational torsion) is uniformly bounded in the fibres E _s , s∈S(k). In this paper, we extend this result in the p-primary case to arbitrary abelian schemes over curves.     

Any nilpotence degree occurs in mod-<Emphasis Type="Italic">p</Emphasis> cohomology rings of <Emphasis Type="Italic">p</Emphasis>-groups

Let p be an odd prime number and let n be an arbitrary positive integer. We prove that there exists a p-group whose mod-p cohomology ring has a nilpotent element H²() satisfying ⁿ0,^n+p–1=0. As a corollary, we exhibit a p-group whose mod-p cohomology ring contains an element of nilpotency degree n+1.Mathematical Subject Classification (2000): 20J06, 20D15, 55R40To Phuong and Nin     

On <Emphasis Type="Italic">CEP</Emphasis>-subgroups of <Emphasis Type="Italic">n</Emphasis>-periodic products

There is a well-known fact, that any group G ₁ is a CEP-subgroup both for the direct product G ₁ × G ₂ and the free productG ₁ * G ₂ of G ₁ with any group G ₂. The paper gives a necessary and sufficient condition providing that a multiplier G _i of a n-periodic product Π _i∈I ⁿ G _i of any family of groups {G _i } _i∈I is a CEP-subgroup. Particularly, the found criterionmeans that any group G ₁ of odd period n ≥ 665 is a CEP-subgroup of the n-periodic product Π _i∈I ⁿ G _i for any group G ₂.     

<Emphasis Type="Italic">c</Emphasis>*-Supplemented subgroups and <Emphasis Type="Italic">p</Emphasis>-nilpotency of finite groups

A subgroup H of a finite group G is said to be c*-supplemented in G if there exists a subgroup K such that G = HK and H ⋂ K is permutable in G. It is proved that a finite group G that is S ₄-free is p-nilpotent if N _G (P) is p-nilpotent and, for all x ∈ G\N _G (P), every minimal subgroup of is c*-supplemented in P and (if p = 2) one of the following conditions is satisfied: (a) every cyclic subgroup of of order 4 is c*-supplemented in P, (b) , (c) P is quaternion-free, where P a Sylow p-subgroup of G and is the p-nilpotent residual of G. This extends and improves some known results. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1011–1019, August, 2007.     

Cyclotomic Hecke Algebras of <Emphasis Type="Italic">G</Emphasis>(<Emphasis Type="Italic">r</Emphasis>, <Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)

In this note, we find a monomial basis of the cyclotomic Hecke algebra \({\mathcal{H}_{r,p,n}}\) of G(r,p,n) and show that the Ariki-Koike algebra \({\mathcal{H}_{r,n}}\) is a free module over \({\mathcal{H}_{r,p,n}}\), using the Gröbner-Shirshov basis theory. For each irreducible representation of \({\mathcal{H}_{r,p,n}}\), we give a polynomial basis consisting of linear combinations of the monomials corresponding to cozy tableaux of a given shape.     

Ergodic polynomials on subsets of <Emphasis Type="Italic">p</Emphasis>-adic integers

We establish results concerning ergodicity on compact subsets of Z _p and study ergodicity of polynomials on subsets of Z₂ and Z₃.     

On <Emphasis Type="Italic">s</Emphasis>-semipermutable subgroups of finite groups and <Emphasis Type="Italic">p</Emphasis>-nilpotency

A subgroup H of a group is said to be s-semipermutable in G if it is permutable with every Sylow p-subgroup of G with (p, |H|) = 1. Using the concept of s-semipermutable subgroups, some new characterizations of p-nilpotent groups are obtained and several results are generalized.     

Irreducible <Emphasis Type="Italic">p</Emphasis>-Brauer characters of <Emphasis Type="Italic">p</Emphasis>-power degree for the symmetric and alternating groups

Starting from the question when all irreducible p-Brauer characters for a symmetric or an alternating group are of p-power degree, we classify the p-modular irreducible representations of p-power dimension in some families of representations for these groups. In particular, this then allows to confirm a conjecture by W. Willems for the alternating groups. Received: 14 June 2006     

<Emphasis Type="Italic">M</Emphasis> <Subscript> <Emphasis Type="Italic">p</Emphasis> </Subscript>-supplemented subgroups of finite groups

A subgroup K of G is M_p-supplemented in G if there exists a subgroup B of G such that G = KB and TB < G for every maximal subgroup T of K with |K: T| = p^α. In this paper we prove the following: Let p be a prime divisor of |G| and let H be ap-nilpotent subgroup having a Sylow p-subgroup of G. Suppose that H has a subgroup D with D_p ≠ 1 and |H: D| = p_α. Then G is p-nilpotent if and only if every subgroup T of H with |T| = |D| is M_p-supplemented in G and N_G(T_p)/C_G(T_p) is a p-group.     

Free-by-Demushkin pro-<Emphasis Type="Italic">p</Emphasis> groups

We give an example of a short exact sequence 1NGD1 of pro-p groups such that the cohomological dimension cd(G)=2, G is (topologically) finitely generated, N is a free pro-p group of infinite rank, D is a Demushkin group, for every closed subgroup S of G containing N and any natural number n the inflation map is an isomorphism but G is not a free pro-p product of a free pro-p group by a Demushkin group. This is a group theoretic version of a question raised by T. Würfel for some special Galois groups.Both authors are partially supported by bolsa de produtividade de pesquisa from CNPq, Brazil and CNPq grant 470272/2003-1.     

Automorphisms of Coxeter groups of type <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

A Coxeter system (W, S) is said to be of type K _n if the associated Coxeter graph Γ_S is complete on n vertices and has only odd edge labels. If W satisfies either of: (1) n = 3; (2) W is rigid; then the automorphism group of W is generated by the inner automorphisms of W and any automorphisms induced by Γ_S. Indeed, Aut(W) is the semidirect product of Inn(W) and the group of diagram automorphisms, and furthermore W is strongly rigid. We also show that if W is a Coxeter group of type K _n then W has exactly one conjugacy class of involutions and hence Aut(W) = Spec(W).