Odd Exponent Research Articles

THE GROUPS OF AUTOMORPHISMS ARE COMPLETE FOR FREE BURNSIDE GROUPS OF ODD EXPONENTS n ≥ 1003

It is proved that the group of automorphisms Aut (B(m, n)) of the free Burnside group B(m, n) is complete for every odd exponent n ≥ 1003 and for any m > 1, that is, it has a trivial center and any automorphism of Aut (B(m, n)) is inner. Thus, the automorphism tower problem for groups B(m, n) is solved and it is showed that it is as short as the automorphism tower of the absolutely free groups. Moreover, the group of all inner automorphisms Inn (B(m, n)) is the unique normal subgroup in Aut (B(m, n)) among all its subgroups, which are isomorphic to free Burnside group B(s, n) of some rank s.

Erratum to: A new proof of the odd primary homotopy exponent of spheres

≥ 3. However, this made critical use of [3, Lemma 2.3], which Gray andRichter later identified as being incorrect without additional assumptions. It maybe possible that these additional assumptions can be shown to hold for the specificapplication of [3, Lemma 2.3] used in the argument of [3, Theorem 1.1], but thisis unclear. Instead, Richter [2] uses different methods to give a proof of Gray’sconjecture which holds for all primes

A Bound on the Order of Non-Abelian Tensor Square of a Prime-Power Group

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.

A new proof of the odd primary homotopy exponent of spheres

We give a different proof of Cohen, Moore and Neisendorfer’s theorem stating that for odd primes p the homotopy exponent of S2n+1 is pn. This is done using methods recently introduced by Gray and the author to give a new construction of Anick’s fibration.

Seismic Reliability Analysis of MDOF Systems Based on Odd Exponent Wavelet

This study proposed non-failure or non-disabled probability of structure suffering random dynamic load within given time. A wavelet method based on its conception was proposed for seismic dynamic reliability analysis of MDOF system：From the Duhamel integral of the dynamic structural response, the structure stochastic response was expressed as earthquake ground motion’s wavelet transform. The ground motion and structure responses were modeled as non-stationary random process using odd exponent wavelets. The first-passage failure criterion and maximum probability were employed to estimate the structural dynamic reliability. The primary advantage of the proposed method is that it does not need to calculate spectral moments from the power density function integral, which are usually difficult to obtain the analytical expressions.

Fixed-point free action of an abelian group of odd non-squarefree exponent

AbstractLet A be a finite group acting fixed-point freely on a finite (solvable) group G. A longstanding conjecture is that if (|G|, |A|) = 1, then the Fitting length of G is bounded by the length of the longest chain of subgroups of A. It is expected that the conjecture is true when the coprimeness condition is replaced by the assumption that A is nilpotent. We establish the conjecture without the coprimeness condition in the case where A is an abelian group whose order is a product of three odd primes and where the Sylow 2-subgroups of G are abelian.

Once more on periodic products of groups and on a problem of A. I. Mal’tsev

A new operation of product of groups, the n-periodic product of groups for odd exponent n ≥ 665, was proposed by the author in 1976 in the paper [1]. This operation is described on the basis of the Novikov-Adyan theory introduced in the monograph [2] of the author. It differs from the classic operations of direct and free products of groups, but has all of the natural properties of these operations, including the so-called hereditary property for subgroups. Thus, the well-known problem of A. I. Mal’tsev on the existence of such new operations was solved. Unfortunately, in the paper [1], the case where the initial groups contain involutions, was not analyzed in detail. It is shown that, in the case where the initial groups contain involutions, this small gap is easily removed by an additional restriction on the choice of defining relations for the periodic product. It suffices to simply exclude products of two involutions of previous ranks from the inductive process of defining new relations for any given rank α. It is suggested that the adequacy of the given restriction follows easily from the proof of the key Lemma II.5.21 in the monograph [2]. We also mention that, with this additional restriction, all the properties of the periodic product given in [1] remain true with obvious corrections of their formulation. Moreover, under this restriction, one can consider n-periodic products for any period n ≥ 665, including even periods.

UNIQUELY 2-DIVISIBLE BOL LOOPS

Although any finite Bol loop of odd prime exponent is solvable, we show there exist such Bol loops with trivial center. We also construct finitely generated, infinite, simple Bruck loops of odd prime exponent for sufficiently large primes. This shows that the Burnside problem for Bruck loops has a negative answer.

On subgroups of free Burnside groups of odd exponent

We prove that for any odd number , every non-cyclic subgroup of the 2-generator free Burnside group of exponent contains a subgroup isomorphic to the free Burnside group of exponent and infinite rank. Various families of relatively free -periodic subgroups are constructed in free periodic groups of odd exponent . For the same groups, we describe a monomorphism such that a word is an elementary period of rank if and only if its image is an elementary period of rank .

Uniform non-amenability of free Burnside groups

The aim of this paper is to show that free Burnside groups of sufficiently large odd exponent are non-amenable in a certain strong sense. More precisely their left regular representations are isolated from the trivial representation uniformly on finite generating sets. It follows that free Burnside groups are of uniform exponential growth. This answers a question of de la Harpe.

NORMAL AUTOMORPHISMS OF FREE BURNSIDE GROUPS OF LARGE ODD EXPONENTS

Let [Formula: see text] be a free Burnside group of a sufficiently large odd exponent n with a basis [Formula: see text] of cardinality at least 2. We prove that every normal automorphism of [Formula: see text] is inner. We also prove that a free Burnside group of large odd exponent n can be normally embedded into group of exponent n only as a direct factor.

On the class groups of cyclotomic extensions in presence of a solution to Catalan's equation

Catalan's conjecture states that the equation x p − y q = 1 has no other integer solutions but 3 2 − 2 3 = 1 . We investigate the consequences of existence of further solutions (with odd prime exponents p , q ) upon the relative class group of the pth cyclotomic extension. We thus obtain several new results which merge into the condition q ≢ 1 mod p and p ≢ 1 mod q . This condition is used in the proof of Catalan's conjecture.

Embedding Free Burnside Groups in Finitely Presented Groups

We construct an embedding of a free Burnside group B(m,n) of odd exponent n > 248 and rank m >1 in a finitely presented group with some special properties. The main application of this embedding is an easy construction of finitely presented nonamenable groups without noncyclic free subgroups (which provides a new finitely presented counterexample to the von Neumann problem on amenable groups). As another application, we construct weakly finitely presented groups of odd exponent n ≫ 1 which are not locally finite.

Uniform non-amenability

For any finitely generated group G an invariant Føl G ⩾ 0 is introduced which measures the “amount of non-amenability” of G. If G is amenable, then Føl G = 0 . If Føl G > 0 , we call G uniformly non-amenable. We study the basic properties of this invariant; for example, its behaviour when passing to subgroups and quotients of G. We prove that the following classes of groups are uniformly non-amenable: non-abelian free groups, non-elementary word-hyperbolic groups, large groups, free Burnside groups of large enough odd exponent, and groups acting acylindrically on a tree. Uniform non-amenability implies uniform exponential growth. We also exhibit a family of non-amenable groups (in particular including all non-solvable Baumslag–Solitar groups) which are not uniformly non-amenable, that is, they satisfy Føl G = 0 . Finally, we derive a relation between our uniform Følner constant and the uniform Kazhdan constant with respect to the left regular representation of G.

Elliptic-Cylindrical Wavelets: The Mathieu Wavelets

This note introduces a new family of wavelets and a multiresolution analysis that exploits the relationship between analyzing filters and Floquet's solution of Mathieu differential equations. The transfer function of both the detail and the smoothing filter is related to the solution of a Mathieu equation of the odd characteristic exponent. The number of notches of these filters can be easily designed. Wavelets derived by this method have potential application in the fields of optics and electromagnetism.

On subgroups of free Burnside groups of large odd exponent

We prove that every noncyclic subgroup of a free $m$-generator Burnside group $B(m,n)$ of odd exponent $n \gg 1$ contains a subgroup $H$ isomorphic to a free Burnside group $B(\infty,n)$ of exponent $n$ and countably infinite rank such that for every normal subgroup $K$ of $H$ the normal closure $ ^{B(m,n)}$ of $K$ in $B(m,n)$ meets $H$ in $K$. This implies that every noncyclic subgroup of $B(m,n)$ is SQ-universal in the class of groups of exponent $n$.

Euler to the Rescue: 10869

(b) n 1 mod 4. Now n 1 is even and n(n + 1)/2 is odd; suppose that both are perfect. By Lemma 1, n 1 = 2p-1(2p 1), where p is prime and greater than 2. Thus n(n + 1)/2 = (22p-1 2p-1 + 1)(22p-2 2p-2 + 1). These two factors are relatively prime. Lemma 2 now requires that one of them be a square. However, 22p-1 2p-1 + 1 -2 mod 3 since p is odd, and also (2p-1 1)2 < 22p-2 2p-2 + 1 < (2p-1)2. Thus neither factor is a square, and n (n + 1)/2 is not perfect. (c) n -_ 2 mod 4. Now n + 1 3 mod 4. Thus some prime q congruent to 3 modulo 4 has odd exponent in the prime factorization of n + 1. Since n and n + 1 are relatively prime, q has the same odd exponent in the prime factorization of n(n + 1)/2. By Lemma 2, a prime congruent to 3 modulo 4 cannot appear with odd exponent in the prime factorization of an odd perfect number. Hence n(n + 1)/2 is not perfect. (d) n =3 mod 4. Now n 1 is divisible by, 2 but not by 4. By Lemma 1, 6 is the only even perfect number not divisible by 4. Thus in this case n 1 is perfect only when n = 7. In this case n(n + 1)/2 equals 28, which is also perfect.

CEP-Subgroups of Free Burnside Groups of Large Odd Exponents

For sufficiently large odd exponent n we construct a CEP-subgroup isomorphic to a free Burnside group B(∞, n) with infinite number of generators in the free Burnside group B(2, n) on two generators. This implies that B(2, n) is SQ-universal inside the Burnside variety.

On 2-groups with submultiplicative spectrum

A group G of complex square invertible matrices is said to have submultiplicative spectrum if for any G, H in G we have that σ( GH)⊂ σ( G) σ( H), where σ( G) denotes the spectrum of G. The question whether such a group can be irreducible arises in the study of the structure of matrix semigroups and is answered in the positive for groups with odd exponent (cf. Radjavi and Rosenthal, Simultaneous Triangularization, Springer, New York, 1999). Here we consider the most important remaining case, 2-groups with submultiplicative spectrum. If there is an element in the group of the kind whose determinant has order equal to either the exponent or half of the exponent of the group, then the group is reducible. We give an example of an irreducible group of the kind such that the order of determinant of every element equals one quarter of the exponent of the group.

Maximum Modulus Sets and Segre Convexity

Let E be a totally real, analytic, n-dimensional manifold, foliated by analytic interpolation submanifolds of codimension 1, in the analytic boundary of a Segre-convex domain in ℂn. Given a canonical defining function of the boundary of Ω in a point 0 of E : Im z1 +R[Re z1, z′, z′]=0. If all the odd exponents in the decomposition of R in irreducible factors, at 0,are greater than 1 then R≥0 and E is locally contained in a maximum modulus set.