Virtually free pro-<Emphasis Type="Italic">p</Emphasis> products

It is shown that a finitely generated pro-p group G which is a virtually free pro-p product splits either as a free pro-p product with amalgamation or as a pro-p HNN-extension over a finite p-group. More precisely, G is the pro-p fundamental group of a finite graph of finitely generated pro-p groups with finite edge groups. This generalizes previous results of W. Herfort and the second author (cf. [2]).     

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]).     

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.     

<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.     

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     

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.     

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.     

Kernels and <Emphasis Type="Italic">p</Emphasis>-Kernels of <Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript>-ary 1-Perfect Codes

The rank of a q-ary code C is the dimension of the subspace spanned by C. The kernel of a q-ary code C of length n can be defined as the set of all translations leaving C invariant. Some relations between the rank and the dimension of the kernel of q-ary 1-perfect codes, over as well as over the prime field , are established. Q-ary 1-perfect codes of length n=(q^m − 1)/(q − 1) with different kernel dimensions using switching constructions are constructed and some upper and lower bounds for the dimension of the kernel, once the rank is given, are established.Communicated by: I.F. Blake     

Cyclic Functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(<Emphasis Type="Bold">R</Emphasis>), 1 \le <Emphasis Type="Italic">p</Emphasis> < \infty

Functions whose translates span L ^p (R) are called L ^p-cyclic functions. For a fixed p \memb [1, \infty], we construct Schwartz-class functions which are L ^r -cyclic for r > p and not L ^r - cyclic for r \le p. We then construct Schwartz-class functions which are L ^r -cyclic for r \ge p and not L ^r -cyclic for r < p. The constructions differ for p \memb (1, 2) and p > 2.     

Approximation of sine-shaped functions by constants in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>, <Emphasis Type="Italic">p</Emphasis> < 1

We investigate the best approximations of sine-shaped functions by constants in the spaces L_p for p < 1. In particular, we find the best approximation of perfect Euler splines by constants in the spaces L_p for certain p(0,1).Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 6, pp. 745–762, June, 2004.     

A note on <Emphasis Type="Italic">n</Emphasis>! modulo <Emphasis Type="Italic">p</Emphasis>

Let p be a prime, \(\varepsilon >0\) and \(0<L+1<L+N < p\). We prove that if \(p^{1/2+\varepsilon }< N <p^{1-\varepsilon }\), then

$$\begin{aligned} \#\{n!\,\,({\mathrm{mod}} \,p);\,\, L+1\le n\le L+N\} > c (N\log N)^{1/2},\,\, c=c(\varepsilon )>0. \end{aligned}$$

We use this bound to show that any \(\lambda \not \equiv 0\ ({\mathrm{mod}}\, p)\) can be represented in the form \(\lambda \equiv n_1!\cdots n_7!\ ({\mathrm{mod}}\, p)\), where \(n_i=o(p^{11/12})\). This refines the previously known range for \(n_i\).

     

Lower semicontinuity and relaxation for integral functionals with <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)- and <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">x,u</Emphasis>)-growth

We consider the questions of lower semicontinuity and relaxation for the integral functionals satisfying the p(x)- and p(x, u)-growth conditions. Presently these functionals are actively studied in the theory of elliptic and parabolic problems and in the framework of the calculus of variations. The theory we present rests on the following results: the remarkable result of Kristensen on the characterization of homogeneous p-gradient Young measures by their summability; the earlier result of Zhang on approximating gradient Young measures with compact support; the result of Zhikov on the density in energy of regular functions for integrands with p(x)-growth; on the author’s approach to Young measures as measurable functions with values in a metric space whose metric has integral representation.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> discrepancy of generalized two-dimensional Hammersley point sets

We determine the L _p discrepancy of the two-dimensional Hammersley point set in base b. These formulas show that the L _p discrepancy of the Hammersley point set is not of best possible order with respect to the general (best possible) lower bound on L _p discrepancies due to Roth and Schmidt. To overcome this disadvantage we introduce permutations in the construction of the Hammersley point set and show that there always exist permutations such that the L _p discrepancy of the generalized Hammersley point set is of best possible order. For the L ₂ discrepancy such permutations are given explicitly. F.P. is supported by the Austrian Science Foundation (FWF), Project S9609, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.     

Automorphisms of Sylow <Emphasis Type="Italic">p</Emphasis>-subgroups of Chevalley groups over <Emphasis Type="Italic">p</Emphasis>-primary residue rings of integers

In this paper, we prove that any automorphism of a Sylow p-subgroup of the Chevalley group over the ring (where p is prime and m ≥ 1) is a product of graph, inner, diagonal, and hypercentral automorphisms. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 8, pp. 121–158, 2006.     

Spin-One <Emphasis Type="Italic">p</Emphasis>-Spin Glass: Exact Solution for Large <Emphasis Type="Italic">p</Emphasis>

We study the low-temperature properties of the p-spin spin glass model in the spin-one (three-state) case for large values of p. We show that the one-step replica symmetry-breaking phase is unstable at a very low temperature, and we calculate the explicit boundary of the stability interval, the Gardner temperature, analytically for large values of p. This temperature for the spin-one model has the same form of dependence on p as in the case of Ising spins (two states). In the one-step replica symmetrybreaking state, a quadrupolar orientational glass coexists with the spin glass and also with a regular quadrupole ordering.     

Continuous <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">p</Emphasis></Subscript>-symmetric distributions

For p > 0, the l _n,p -generalized surface measure on the l _n,p -unit sphere is studied and used for deriving a geometric measure representation for l _n,p -symmetric distributions having a density.     

Splitting theorems for pro-<Emphasis Type="Italic">p</Emphasis> groups acting on pro-<Emphasis Type="Italic">p</Emphasis> trees

Let G be an infinite finitely generated pro-p group acting on a pro-p tree such that the restriction of the action to some open subgroup is free. We prove that G splits over an edge stabilizer either as an amalgamated free pro-p product or as a pro-p \({\text {HNN}}\)-extension. Using this result, we prove under a certain condition that free pro-p products with procyclic amalgamation inherit from its amalgamated free factors the property of each 2-generated pro-p subgroup being free pro-p. This generalizes known pro-p results, as well as some pro-p analogues of classical results in abstract combinatorial group theory.