Gluing endo-permutation modules

Large subgroups of small class in finite p-groups

Centralizers of elementary Abelian subgroups in finite p-groups

If 0 → A → C → B → 0 is an exact sequence of abelian groups, if ƒ is a 2-cocyle for this extension, if α ∈ End A, and if β ∈ End B, then a necessary and sufficient condition that α extend to an endomorphism γ of C which induces β is that (M) αƒ and ƒβ be cohomologous ; see Montgomery (2). We shall extend this result to the case where 1 → A → G → B → 1 is an exact sequence of groups and A is abelian.

Given a long exact sequence of abelian groups [fórmula disponible al document original] a short exact sequence of complexes of free abelian groups is constructed whose cohomology long exact sequence is precisely L. In this sense, L is realized . Two techniques which are introduced to reduce or replace lengthy diagram chasing arguments may be of interest to some readers. One is an arithmetic of bicartesian squares; the other is the use of the fact that categories of morphisms of abelian categories are themselves abelian.

Borel–Smith functions and the Dade group

A remark on the Dade group and the Burnside group

We define the notion of the Dade group of a fusion system and show that some of the gluing and detection results for Dade groups of finite p-groups due to Bouc and Thévenaz in [S. Bouc and J. Thévenaz. Gluing torsion endo-permutation modules. (Preprint.)], [S. Bouc and J. Thévenaz. A sectional characterization of the Dade group. J. Group Theory11 (2008), 155–183.] extend to Dade groups of fusion systems.

Nonnegative nilpotent lower triangular completions of a nonnegative nilpotent matrix are studied. It is shown that for every natural number between the index of the matrix and its order, there exists a completion that has this number as its index. A similar result is obtained for the rank. However, unlike the case of complex completions of complex matrices, it is proved that for every nonincreasing sequence of nonnegative integers whose sum is n, there exists an n n nonnegative nilpotent matrix A such that for every nonnegative nilpotent lower triangular completion, B, of A, B 6= A, ind(B) > ind(A). AMS(MOS) subject classi cation. 15A21, 15A48

Finite p-groups with many minimal nonabelian subgroups

Let G be a group and mathrm{IA}(G) denote the group of all automorphisms of G, which induce identity map on the abelianized group G_{ab}=G/G'. Also the group of those mathrm{IA}-automorphisms which fix the centre element-wise is denoted by mathrm{IA_Z}(G). In the present article, among other results and under some condition we prove that the derived subgroups of finite p-groups, for which mathrm{IA_Z}-automorphisms are the same as central automorphisms, are either cyclic or elementary abelian.

In this article, we prove that the Schur multiplier of a finite 𝑝-group of maximal class of order p n p^{n} ( 4 ≤ n ≤ p + 1 4\leq n\leq p+1 ) is elementary abelian. The case n = p + 1 n=p+1 settles a question raised by Primož Moravec in an earlier article.

This article improves on an upper bound for the order of the non-abelian tensor square of a finite p-group G given in [3]. In particular, applying this for finite p-groups of order p n with factor group G/G′ of order p m , the bound p nm attains if and only if G is elementary abelian of rank n, quaternion group of order 8, or extra special p-group of order p 3 with odd exponent p.

Generalizing a celebrated theorem of Thompson, R. P. Martineau has established [4; 5] that a finite group which admits an elementary abelian group of automorphisms with trivial fixed-point subgroup is necessarily solvable. A critical observation in his approach to this problem is the fact that, corresponding to each prime divisor of its order, such a group contains a unique Sylow subgroup invariant (as a set) under the action. Hence, the theorem we shall derive here represents a modest extension of Martineau's result.

We prove some inequalities which follow from the log-convexity of the sequence of Motzkin numbers Mn and from the log-concavity of the sequence Mn n! . Mathematics subject classification (2000): 05A20, 26D99. Keywordsandphrases: Inequalities,Motzkinnumbers, log-convexity, log-concavity, integer sequences. RE F ER EN C ES [1] M. AIGNER, Motzkin numbers, European Journal of Combinatorics 19 (1998), 663–675. [2] D. CALLAN, Notes on Motzkin and Schroeder Numbers, preprint. [3] R. DONAGHEY AND L. W. SHAPIRO, Motzkin Numbers, Journal of Combinatorial Theory series A 23 (1977), 291–301. [4] T. DOSLIC, D. SVRTAN AND D. VELJAN, Secondary Structures, submitted. [5] T. MOTZKIN, Relation between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanental preponderance and for non-associative products, Bulletin of American Mathematical Society 54 (1948), 352–360. [6] R. STANLEY, Enumerative Combinatorics II, Cambridge Univ. Press, Cambridge, 1999. [7] D. VELJAN, Combinatorial and Discrete Mathematics, Algoritam, Zagreb, 2001 (in Croatian). [8] P. R. STEIN AND M. WATERMAN, On Some New Sequences Generalizing the Catalan and the Motzkin Numbers, Discr. Math. 26 (1978), 261–272. c © , Zagreb Paper MIA-05-18 Mathematical Inequalities & Applications www.ele-math.com mia@ele-math.com

We prove here that a nonabelian finite p-group G has exactly one maximal subgroup with a noncyclic center if and only if Z(G) is cyclic and G has exactly one normal abelian subgroup of type (p, p).