P-groups Of Class Research Articles

Embeddings of minimal non-abelian p-groups

Abstract By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. For a given finite group G, let p(G) denote the minimal degree of a faithful representation of G by permutation matrices, and let c(G) denote the minimal degree of a faithful representation of G by quasi-permutation matrices. See [4] . It is easy to see that c(G) is a lower bound for p(G). Behravesh [H. Behravesh, The minimal degree of a faithful quasi-permutation representation of an abelian group, Glasg. Math. J. 39 (1) (1997) 51–57] determined c(G) for every finite abelian group G and also [H. Behravesh, Quasi-permutation representations of p-groups of class 2, J. Lond. Math. Soc. (2) 55 (2) (1997) 251–260] gave the algorithm of c(G) for each finite group G. In this paper, we first improve this algorithm and then determine c(G) and p(G) for an arbitrary minimal non-abelian p-group G.

Finite p-Groups of Class 3 All of Whose Proper Sections Have Class at Most 2

In this paper, we give a characterization and a complete classification of minimal-class-3 p-groups, that is, the p-groups of class 3 all of whose proper sections have class at most 2.

A Characterization of Groups Whose Lattices of Subgroups are n–Mp+1 Chains for All Primes p

Whitman, P.M. and Birkhoff, G. answered a well-known open question that for each lattice L thereexists a group G such that L can be embedded into the lattice Sub(G) of all subgroups of G. Gratzer, G. hascharacterized that G is a finite cyclic group if and only if Sub(G) is a finite distributive lattice. Ratanaprasert, C.and Chantasartrassmee, A. extended a similar result to a subclass of modular lattices M m by characterizing allintegers m 3 such that there exists a group G whose Sub(G) is isomorphic to Mm and also have characterizedall groups G whose Sub(G) is isomorphic to M m for some integers m. On the other hand, a very well-known openquestion in Group Theory asked for the number of all subgroups of a group. In this paper, we consider theextension of the subclass M m for all integers m ³ 3 of modular lattices, the class of n–M p+1 chains for all primesp, and all n ³ 1 and characterized all groups G whose Sub(G) is an n–M p+1 chain. It happens that G is a groupwhose Sub(G) is an n–M p+1 chain if and only if G is an abelian p-group of the form pn p Z ´ Z . Moreover, wecan tell numbers of all subgroups of order p i for each 1 i n of the special class of p-groups. Key Words: Modular lattice; Lattice of subgroups; p-group

Fusions of Character Tables and Schur Rings of Abelian Groups

If the character table of a finite group H satisfies certain conditions, then the classes and characters of H can fuse to give the character table of a group G of the same order. We investigate the case where H is an abelian group. The theory is developed in terms of the S-rings of Schur and Wielandt. We discuss certain classes of p-groups which fuse from abelian groups and give examples of such groups which do not. We also show that a large class of simple groups do not fuse from abelian groups. The methods to show fusion include the use of extensions which are Camina pairs, but other techniques on S-rings are also developed.

On the vanishing of subspaces of alternating bilinear forms

Given a field F and integer n≥3, we introduce an invariant sn (F) which is defined by examining the vanishing of subspaces of alternating bilinear forms on 2-dimensional subspaces of vector spaces. This invariant arises when we calculate the largest dimension of a subspace of n × n skew-symmetric matrices over F which contains no elements of rank 2. We show how to calculate sn (F) for various families of field F, including finite fields. We also prove the existence of large subgroups of the commutator subgroup of certain p-groups of class 2 which contain no non-identity commutators.

Characters and Representations of p-Groups of Class 2

In this paper, a systematic way is described for the construction of all irreducible characters and irreducible representations of finite p-groups of class 2.

On Modified Circular Units and Annihilation of real Classes

For an abelian totally real number field F and an odd prime number p which splits totally in F, we present a functorial approach to special “p-units” previously built by D. Solomon using “wild” Euler systems. This allows us to prove a conjecture of Solomon on the annihilation of the p-class group of F (in the particular context here), as well as related annihilation results and index formulae.

𝑝-class groups of certain extensions of degree 𝑝

Let p p be an odd prime number. In this article we study the distribution of p p -class groups of cyclic number fields of degree p p , and of cyclic extensions of degree p p of an imaginary quadratic field whose class number is coprime to p p . We formulate a heuristic principle predicting the distribution of the p p -class groups as Galois modules, which is analogous to the Cohen-Lenstra heuristics concerning the prime-to- p p -part of the class group, although in our case we have to fix the number of primes that ramify in the extensions considered. Using results of Gerth we are able to prove part of this conjecture. Furthermore, we present some numerical evidence for the conjecture.

A note on finite semifields and certain p-groups of class 2

In this paper the authors discuss a relationship between finite semifields of characteristic p and certain finite p-groups of nilpotence class 2.

Nilpotency degree of integral cohomology classes of p-groups

Let p be a prime number and let G be a p-group which is not elementary abelian. For every integral cohomology class \( \xi \) of G which restricts trivially to all proper subgroups, we show that \( \xi^{p} = 0 \) if p > 2 or \( \textrm{deg}(\xi) \) is even, and \( \xi^{3} = 0 \) if p = 2 and \( \textrm{deg}(\xi) \) is odd. This result is applied to get an upper bound, which is \( \frac{|G|}{p} \), for the nilpotence degrees of nilpotent integral cohomology classes of G.

Galois co-descent for étale wild kernels and capitulation

Let F be a number field with ring of integers o F . For a fixed prime number p and i≥2 the étale wild kernels WK 2i-2 e ´t (F) are defined as kernels of certain localization maps on the i-fold twist of the p-adic étale cohomology groups of spec o F [1 p]. These groups are finite and coincide for i=2 with the p-part of the classical wild kernel WK 2 (F). They play a role similar to the p-part of the p-class group of F. For class groups, Galois co-descent in a cyclic extension L/F is described by the ambiguous class formula given by genus theory. In this formula, the only factor which is not well mastered is the norm index [U F ′ :U F ′ ∩N L/F (L * )] for the p-units U F ′ . The aim of this paper is the study of the Galois co-descent for wild kernels: Given a cyclic extension L/F of degree p with Galois group G, we show that the transfer map WK 2i-2 e ´t (L) G →WK 2i-2 e ´t (F) is onto except in a very special case, then we determine its kernel as the cokernel of a certain cup-product with values in a Brauer group. This approach also yields a genus formula, analogous to the one for class groups, comparing the sizes of WK 2i-2 e ´t (L) G and WK 2i-2 e ´t (F) where p-units U F ′ are replaced by odd K-theory groups. When p is odd, we illustrate the method by finding all Galois p-extensions of Q, for which the p-part of the classical wild kernel is trivial. For p≥5,they turn out to be the layers of the cyclotomic Z p -extension of Q.

Character Degrees, Class Sizes, and Normal Subgroups of a Certain Class of p-Groups

Character Degrees, Class Sizes, and Normal Subgroups of a Certain Class of p-Groups

Integral and modular representations attached to the Selmer group of an elliptic curve with complex multiplication

Let E be an elliptic curve with complex multiplication over the ring of integers of an imaginary quadratic field K. Denote by p an odd prime that splits into $\mathfrak{p}\overline{\mathfrak{p}}$ in $K/\Bbb Q$ and by $K_\infty$ the unique $\mathbb{Z}$ -extension of K totally ramified above $\mathfrak{p}$ . It is well-known that the Selmer group attached to any finite extension $F_{\infty }$ of $K_{\infty }$ is analogous to the minus part of the p-class group of divisors of the cyclotomic $\mathbb{Z}$ - extensions of CM number fields. One of the most striking examples of this analogy is the existence of a translation formula a la Kida for the codimension of the Selmer group at the top of the tower. In this article we carry on the analogy with the presentations of results similar to those proven by Gold and Madan in the cyclotomic case (see [8]), which were the continuation of Kida's work. More precisely, we describe the $\mathbb{Z}[G]$ -structure of the Selmer group when G is a cyclic group of order p or $p^2$ . In addition, we study the modular representation of G on the subgroup of points of order p of the Selmer group, when G is cyclic of order $p^n$ .

Torsion-free extensions of butler groups by primary groups

The class of those abelian p-groups T is characterized for which every torsion-free group G with T= G/B is a B2-group whenever the subgroup B is a B2-group. This class of p-groups turns out to be the same as the class obtained by Dugas and Irwin [4] who dealt with the analogous problem for free (rather than B2 )groups. It is also shown that the totally reduced p-groups (defined below) belong to the class of p-groups under consideration

Bilinear and Quadratic Maps, and Some p-Groups of Class 2

Bilinear and Quadratic Maps, and Some p-Groups of Class 2

The Edge-Orbit Conjecture of Babai

This paper proves the Edge-Orbit Conjecture stated by L. Babai (1981, in "Combinatorics" (H. N. V. Temperley, Ed.), pp. 1-40, Cambridge Univ. Press, London). We say a graph X represents a group G if Aut( X) ≅ G. Let m e ( G) be the minimum number of edge orbits among all graphs K which represent G. The Edge-Orbit Conjecture was that m e ( G) is unbounded when G ranges over all finite groups. We show this is true using p-groups of class two and exponent p. The proof uses a characterization theorem from a recent paper of L. Babai, A. J. Goodman, and L. Lovász (1991, European J. Cambin. 12) to bound m e ( G) from below when all subgroups of G are either "small" enough or "large" enough (so that there will be enough automorphisms leaving any small set of subgroups invariant). Then a probabilistic proof is used to show non-constructively that plenty of groups exist whose subgroups have this property. The proof shows that there is an infinite family of groups G for which m e ( G) has a lower bound proportional to [formula].

Sur la structure des groupes de classes relatives. Avec un appendice d'exemples numériques par T. Berthier

Following results by R. Schoof and by H.W. Lenstra–R. Schoof, we give a method allowing to find, for all prime p not dividing [F:ℚ], a system of generators of the relative p-class group of the imaginary abelian field F, only with the knowledge of Bernoulli numbers B 1 (ψ -1 ). Numerical examples are given for p=3 and p=5 about cyclic extensions of degrees 2 and 4. The first example of p-class group having a non cyclic χ-component (for a ℚ p -irreducible odd and not quadratic character χ) has been found by T. Berthier for p=5, and the cyclic quartic field of conductor 37×51 containing ℚ(37).

Class groups of abelian fields, and the main conjecture

This first part of this paper gives a proof of the main conjecture of Iwasawa theory for abelian base fields, including the case p=2, by Kolyvagin’s method of Euler systems. On the way, one obtains a general result on local units modulo circular units. This is then used to deduce theorems on the order of χ-parts of p-class groups of abelian number fields: first for relative class groups of real fields (again including the case p=2). As a consequence, a generalization of the Gras conjecture is stated and proved.

On p-class groups of cyclic extensions of prime degree p of number fields

On p-class groups of cyclic extensions of prime degree p of number fields

Centralizing properties in simple locally finite groups and large finite classical groups

AbstractThe following question is discussed and evidence for and against it is advanced: is it true that if F is an arbitrary finite subgroup of an arbitrary non-linear simple locally finite group G, then CG(F) is infinite? The following points to an affirmative answer.Theorem A. Let F be an arbitrary finite subgroup of a non-linear simple locally finite group G. Then there exist subgroups D ◃ C ≤ G such that F centralizes C/D, F∩C ≤ D, and C/D is a direct product of finite alternating groups of unbounded orders. In particular, F centralizes an infinite section of G.Theorem A is deduced from a “local” version, namelyTheorem B. There exists an integer valued function f(n, r) with the following properties. Let H be a finite group of order at most n, and suppose that H ≤ S, where S is either an alternating group of degree at least f = f(n, r) or a finite simple classical group whose natural projective representation has degree at least f. Then there exist subgroups D ◃ C ≤ S such that (i) [H, C] ≤ D, (ii) H ∩ C ≤ D, (iii) C/D ≅ Alt(r), (iv) D = 1 if S is alternating, and D is a p-group of class at most 2 and exponent dividing p2 if S is a classical group over a field of characteristic p.The natural “local version” of our main question is however definitely false.Proposition C. Let p be a given prime. Then there exists a finite group H that can be embedded in infinitely many groups PSL(n, p) as a subgroup with trivial centralizer.