Lifting group homomorphisms

In this chapter we develop the algebra underlying multirate filter structures over free abelian groups of finite rank. Suppose A is a free abelian group of rank N. Each subgroup Δ of A is a free abelian group of rank n ≤ N. The basis theorem describes how Δ sits in A in the sense that it characterizes the quotient group A/Δ. This quotient group can be a finite abelian group a free abelian group of finite rank the direct product of a finite abelian group and a free abelian group of finite rank.

Let (X,≤) be a partially ordered set (in short, a poset). The automorphism group Aut (X,≤) is the group of all permutations g of X such that x ≤ y if and only if xg ≤ yg for all x,y∈X. We say that (X,≤) is sharply transitive if Aut(X,≤) is sharply transitive on X, that is, for x,y∈X there exists a unique g∈Aut(X,≤) with y = xg. Sharply transitive totally ordered sets have been studied by Ohkuma[4, 5], Glass, Gurevich, Holland and Shelah [3] (see also [2] and [6]). Whereas the only countable sharply transitive totally ordered set is the set of integers, there are a great variety of countable sharply transitive posets. Amongst other results, in [1] the author showed that there are countably many non-isomorphic sharply transitive posets whose automorphism group is infinite cyclic (and also gave a full description of those), whereas there are 2N0 non-isomorphic sharply transitive posets whose automorphism group is isomorphic to the additive group of the rational numbers. This suggests also that one should consider the analogous problem for free abelian groups. The purpose of this note is to show that whenever G is a countable free abelian group then there exists a sharply transitive poset whose automorphism group is isomorphic to G, and that there are already 2N0 non-isomorphic sharply transitive posets whose automorphism group is the free abelian group of rank 2.

The primary purpose of this chapter is the analysis of the group Wq(R) in the case of an arithmetic Dedekind domain R. The characterization of the elements of Br(R)2 as the Brauer classes of Clifford algebras of quadratic spaces over R of rank 4 and trivial Arf invariant is an important step along the way. The quadratic spaces which it provides make it possible to describe Ker Arf by use of the total signature. In the special case where R is the ring of integers in a number field, the quadratic group Qu(R) is closely related to the ideal class group, and this connection together with the description of Ker Arf implies that Wq(R) ≅ Cl(R)2 ⊕ G, where C1(R) is the ideal class group of R and G is a free Abelian group of rank with r the number of real embeddings of the number field. An additional focus is the comparison of the number theory of Wq(R) with that of W(R) and the structure of the quotient W(R)/Wq(R).

I have developed an algorithm and written a GAP functions pautfreegroup:=function(n) (Finite Presentation of automorphism Groups of Free group of Rank n), and autfreeabeliangroup:=function(n) (Finite Presentation of automorphism Groups of Free abelian group of Rank n). Functions using GAP system for computation of a finite presentation for Aut(F_n ) and GL(n,z) respectively. In order to do that we have given a description for the presentation of Aut(F_n) and GL(n,z), the automorphism groups of free groups and free abelian groups respectively.

The structure of the connected component of the identity of the idele-class group of a number field is given in Artin-Tate [1]. It seems worthwhile to investigate whether this result is the trivial of a theorem in the arithmetic of algebraic tori, i.e. the case corresponding to the multiplicative group Gm (as is true of the unit theorem and the finiteness of the class number). It turns out that the formalism of the theory of tori defined over number fields allows one to generalize Artin's proof line-by-line, obtaining the structure of the identity component of the adele-class group of any such torus. I am grateful to Professor T. Ono for suggesting the problem to me. I give now a summary of the definitions and results that I shall need (for all this, see Ono [3]). I denote by Z, Q, R, and C the rings of respectively integers, rationals, real numbers, and complex numbers, and by T an algebraic torus defined over a number field k. T is the group of rational characters of T. If the dimension of T is d, then T is a free abelian group of rank d [3, ?1.1]. T is split by a finite extension K of k, which I choose to be normal over Q. Then the group TK of points of T rational over K is isomorphic to the product (K*)d of K* with itself d times. Every & 'T is a rational map defined over K; hence if g denotes the galois group of K over k, T has the structure of a g-module. The group Hom(Ti, K*) has the structure of a gmodule, given byfo() = (f(a)), and we have g-isomorphisms

Quasi-pure projective and injective torsion free abelian groups of rank 2

On λ-homomorphic skew braces

Let q = p s be a power of a prime number p and let $${\mathbb {F}_{\rm q}}$$ be a finite field with q elements. This paper aims to demonstrate the utility and relation of composed products to other areas such as the factorization of cyclotomic polynomials, construction of irreducible polynomials, and linear recurrence sequences over $${\mathbb {F}_{\rm q}}$$ . In particular we obtain the explicit factorization of the cyclotomic polynomial $${\Phi_{2^nr}}$$ over $${\mathbb {F}_{\rm q}}$$ where both r ? 3 and q are odd, gcd(q, r) = 1, and $${n\in \mathbb{N}}$$ . Previously, only the special cases when r = 1, 3, 5, had been achieved. For this we make the assumption that the explicit factorization of $${\Phi_r}$$ over $${\mathbb {F}_{\rm q}}$$ is given to us as a known. Let $${n = p_1^{e_1}p_2^{e_2}\cdots p_s^{e_s}}$$ be the factorization of $${n \in \mathbb{N}}$$ into powers of distinct primes p i , 1 ? i ? s. In the case that the multiplicative orders of q modulo all these prime powers $${p_i^{e_i}}$$ are pairwise coprime, we show how to obtain the explicit factors of $${\Phi_{n}}$$ from the factors of each $${\Phi_{p_i^{e_i}}}$$ . We also demonstrate how to obtain the factorization of $${\Phi_{mn}}$$ from the factorization of $${\Phi_n}$$ when q is a primitive root modulo m and $${{\rm gcd}(m, n) = {\rm gcd}(\phi(m),{\rm ord}_n(q)) = 1.}$$ Here $${\phi}$$ is the Euler's totient function, and ord n (q) denotes the multiplicative order of q modulo n. Moreover, we present the construction of a new class of irreducible polynomials over $${\mathbb {F}_{\rm q}}$$ and generalize a result due to Varshamov (Soviet Math Dokl 29:334---336, 1984).

Let $q\ge 5$ be a prime and put $q^*=(-1)^{(q-1)/2}\cdot q$. We consider the integer sequence $u_q(1),u_q(2),\ldots,$ with $u_q(j)=(3^j-q^*(-1)^j)/4$. No term in this sequence is repeated and thus for each $n$ there is a smallest integer $m$ such that $u_q(1),\ldots,u_q(n)$ are pairwise incongruent modulo $m$. We write $D_q(n)=m$. The idea of considering the discriminator $D_q(n)$ is due to Browkin (2015) who, in case $3$ is a primitive root modulo $q,$ conjectured that the only values assumed by $D_q(n)$ are powers of $2$ and of $q$. We show that this is true for $n\neq 5$, but false for infinitely many $q$ in case $n=5$. We also determine $D_q(n)$ in case 3 is not a primitive root modulo $q$. Browkin's inspiration for his conjecture came from earlier work of Moree and Zumalacarregui (2016), who determined $D_5(n)$ for $n\ge 1$, thus establishing a conjecture of Salajan. For a fixed prime $q$ their approach is easily generalized, but requires some innovations in order to deal with all primes $q\ge 7$ and all $n\ge 1$. Interestingly enough, Fermat and Mirimanoff primes play a special role in this.

We study the augmentation quotients of the IA-automorphism group of a free group and a free metabelian group. First, for any group G , we construct a lift of the k ‐th Johnson homomorphism of the automorphism group of G to the k ‐th augmentation quotient of the IA-automorphism group of G . Then we study the images of these homomorphisms for the case where G is a free group and a free metabelian group. As a corollary, we detect a Z‐free part in each of the augmentation quotients, which can not be detected by the abelianization of the IA-automorphism group. 20F28; 16S34

Describing the group of units of a group ring is a classical problem. Let [Formula: see text] be a rational prime number. We set [Formula: see text] a primitive root of unity of order [Formula: see text], [Formula: see text] the ring of [Formula: see text]-cyclotomic integers, [Formula: see text] a finite abelian [Formula: see text]-group and [Formula: see text] the group of the units [Formula: see text] of [Formula: see text] such that [Formula: see text], where [Formula: see text] is the augmentation map. We will prove that all the elements of the group [Formula: see text] arise from the units of the group ring [Formula: see text], where [Formula: see text] is the cyclic group of order [Formula: see text]. As an application, we describe explicitly the group of units of the group ring [Formula: see text] when [Formula: see text] is an elementary abelian [Formula: see text]-group and [Formula: see text] is a regular prime number.

Let be the cyclic group of order m and N be the direct product of n copies of . Let Sn be the symmetric group of degree n. The wreath product of with Sn is a split extension of N by Sn, called the generalized symmetric group, here denoted by B(m, n). In his Ph.D. thesis Almestady presented a combinatorial method for constructing the Fischer-Clifford matrices of B(m, n). However as a few examples for small values of m and n show, the manual calculation of these matrices presents formidable problems and hence a computerized approach to this combinatorial method is necessary. In a previous paper the current authors have given a computer programme that computes matrices which are row equivalent to the Fischer-Clifford matrices of B(2, n). Here that programme is generalized to B(m, n), where m is any positive integer. It is anticipated that with some improvements, a number of the programmes given here can be incorporated into GAP. Indeed with further development work these programmes should lead to an alternative method for computing the character table of B(m, n) in GAP.

Using the computational approach introduced in [Agore A.L., Bontea C.G., Militaru G., J. Algebra Appl. 12 (2013), 1250227, 24 pages, arXiv:1207.0411] we classify all coalgebra split extensions of $H_4$ by $k[C_n]$, where $C_n$ is the cyclic group of order $n$ and $H_4$ is Sweedler's $4$-dimensional Hopf algebra. Equivalently, we classify all crossed products of Hopf algebras $H_4 \# k[C_{n}]$ by explicitly computing two classifying objects: the cohomological 'group' ${\mathcal H}^{2} ( k[C_{n}], H_4)$ and $\text{CRP}( k[C_{n}], H_4):=$ the set of types of isomorphisms of all crossed products $H_4 \# k[C_{n}]$. More precisely, all crossed products $H_4 \# k[C_n]$ are described by generators and relations and classified: they are $4n$-dimensional quantum groups $H_{4n, \lambda, t}$, parameterized by the set of all pairs $(\lambda, t)$ consisting of an arbitrary unitary map $t : C_n \to C_2$ and an $n$-th root $\lambda$ of $\pm 1$. As an application, the group of Hopf algebra automorphisms of $H_{4n, \lambda, t}$ is explicitly described.

Long colimits of topological groups II: Free groups and vector spaces

Let p be an odd prime number and $$\ell $$ an odd prime number dividing $$p-1$$ . We denote by $$F=F_{p,\ell }$$ the real abelian field of conductor p and degree $$\ell $$ , and by $$h_F$$ the class number of F. For a prime number $$r \ne p,\,\ell $$ , let $$F_{\infty }$$ be the cyclotomic $$\mathbb {Z}_r$$ -extension over F, and $$M_{\infty }/F_{\infty }$$ the maximal pro-r abelian extension unramified outside r. We prove that $$M_{\infty }$$ coincides with $$F_{\infty }$$ and consequently $$h_F$$ is not divisible by r when r is a primitive root modulo $$\ell $$ and r is smaller than an explicit constant depending on p.