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Fernando Szechtman

Publications and source records attributed to Fernando Szechtman.

At least 19 recordsLinked to original sources
arXiv2026

Substitution groups of formal power series

Let $G$ be the group of power series $x+a_2x^2+a_3x^3+\cdots\in R[[x]]$ under substitution, where $R$ is a commutative ring with $1\neq 0$ of prime characteristic $p$. Given any $n\geq 1$, the subgroup $K_n=\{x+a_{n+1}x^{n+1}+a_{n+2}x^{n+2}+\cdots\,|\, a_i\in R\}$ is normal in $G$, and the quotient $G_n=G/K_n$ is the group of truncated polynomials over $R$ of degree $\leq n$ under substitution. In this paper, we compute the exponent of the image of $K_r$ in $G_n$, for all $r,n\geq 1$, indicating in every case a family of elements realizing this exponent.

math.GR↗
arXiv2026

A family of groups extending McLain's

Given a strict partial order $\Delta$ on a set $\Lambda$ and an arbitrary ring $R$ with $1\neq 0$, the corresponding McLain group $M(\Delta)$ has been studied in depth. We construct a larger family of McLain groups $G(\Delta)$, where $\Delta$ is neither asymmetric nor transitive, while satisfying two weaker axioms. Structural properties common to all members~$G(\Delta)$ of this new family are investigated, including a group presentation, a description of the factors of its descending central series, a canonical form for its elements relative to any total order on~$\Delta$, and a recursive determination of its upper central series. In addition, we prove the natural isomorphism $G(\Delta)/G(\Gamma)\cong G(\Delta\setminus\Gamma)$, where $\Gamma$ is a normal subset $\Gamma$ of $\Delta$, and $G(\Gamma)$ and $G(\Delta\setminus\Gamma)$ are extended McLain groups on their own right. This result has no parallel in the classical context.

math.GR↗
arXiv2025

Artin-Schreier towers of finite fields

Given a prime number $p$, we consider the tower of finite fields $F_p=L_{-1}\subset L_0\subset L_1\subset\cdots$, where each step corresponds to an Artin-Schreier extension of degree $p$, so that for $i\geq 0$, $L_{i}=L_{i-1}[c_{i}]$, where $c_i$ is a root of $X^p-X-a_{i-1}$ and $a_{i-1}=(c_{-1}\cdots c_{i-1})^{p-1}$, with $c_{-1}=1$. We extend and strengthen to arbitrary primes prior work of Popovych for $p=2$ on the multiplicative order of the given generator $c_i$ for $L_i$ over $L_{i-1}$. In particular, for $i\geq 0$, we show that $O(c_i)=O(a_i)$, except only when $p=2$ and $i=1$, and that $O(c_i)$ is equal to the product of the orders of $c_j$ modulo $L_{j-1}^\times$, where $0\leq j\leq i$ if $p$ is odd, and $i\geq 2$ and $1\leq j\leq i$ if $p=2$. We also show that for $i\geq 0$, the $\mathrm{Gal}(L_i/L_{i-1})$-conjugates of $a_i$ form a normal basis of $L_i$ over $L_{i-1}$. In addition, we obtain the minimal polynomial of $c_1$ over $F_p$ in explicit form.

math.NT↗
arXiv2025

Sylow subgroups of the Macdonald group on 2 parameters

Consider the Macdonald group $G(α,β)=\langle A,B\,|\, A^{[A,B]}=A^α,\, B^{[B,A]}=B^β\rangle$, where $α$ and $β$ are integers different from one. We fill a gap in Macdonald's original proof that $G(α,β)$ is nilpotent, and find the order and nilpotency class of each Sylow subgroup of $G(α,β)$.

math.GR↗
arXiv2024

The automorphism group of certain polycyclic groups

For $β\in{\mathbb Z}$, let $G(β)=\langle A,B\,|\, A^{[A,B]}=A,\, B^{[B,A]}=B^β\rangle$ be the infinite Macdonald group, and set $C=[A,B]$. Then $G(β)$ is a nilpotent polycyclic group of the form $\langle A\rangle\ltimes\langle B,C\rangle$, where $A$ has infinite order. If $β\neq 1$, then $G(β)$ is of class 3 and $\langle B,C\rangle$ is a finite metacyclic group of order $|β-1|^3$, which is an extension of $C_{(β-1)^2}$ by $C_{|β-1|}$, split except when $v_2(β-1)=1$, while $G(1)$ is the integral Heisenberg group, of class 2 and $\langle B,C\rangle\cong{\mathbb Z}^2$. We give a full description of the automorphism group of $G(β)$. If $β\neq 1$, then $|\mathrm{Aut}(G(β))|=2(β-1)^4$ and we exhibit an imbedding $\mathrm{Aut}(G(β))\hookrightarrow {\mathrm GL}_4({\mathbb Z}/(β-1){\mathbb Z})$, but for the case $β\in\{-1,3\}$ when 5 is required instead of 4. When $β$ is even the automorphism group of $\langle B,C\rangle$ can be obtained from the work of Bidwell and Curran \cite{BC}, and we indicate which of their automorphisms extend to an automorphism of $G(β)$. In general, we give necessary and sufficient conditions for $G(β)$ to be isomorphic to $G(γ)$. When $\gcd(β-1,6)=1$, we determine the automorphism group of $L(β)=G(β)/\langle A^{β-1}\rangle$, which is a relative holomorph of $\langle B,C\rangle$, and $\langle A^{β-1}\rangle$ is a characteristic subgroup of $G(β)$. The map $\mathrm{Aut}(G(β))\to \mathrm{Aut}(L(β))$ is injective and $\mathrm{Aut}(L(β))$ is an extension of the Heisenberg group over ${\mathbb Z}/(β-1){\mathbb Z}$ direct product $C_{β-1}$, by the holomorph of $C_{β-1}$.

math.GR↗
arXiv2024

Sums of binomial coefficients modulo $p$ and groups of exponent $p^n$

We give a simple matrix-based proof of congruence equations modulo a prime $p$ involving sums of binomial coefficients appearing in Pascal's triangle. These equations can be used to construct some groups of exponent $p^n$. These groups, as well as others of exponent $p^{n+1}$, explain why $p=2$ is not really an exceptional prime in relation to the Heisenberg group over the field with $p$ elements.

math.NT↗
arXiv2024

Isomorphism of relative holomorphs and matrix similarity

Let $V$ be a finite-dimensional vector space over the field with $p$ elements, where $p$ is a prime number. Given arbitrary $α,β\in \mathrm{GL}(V)$, we consider the semidirect products $V\rtimes\langle α\rangle$ and $V\rtimes\langle β\rangle$, and show that if $V\rtimes\langle α\rangle$ and $V\rtimes\langle β\rangle$ are isomorphic, then $α$ must be similar to a power of $β$ that generates the same subgroup as $β$; that is, if $H$ and $K$ are cyclic subgroups of $\mathrm{GL}(V)$ such that $V\rtimes H\cong V\rtimes K$, then $H$ and $K$ must be conjugate subgroups of $\mathrm{GL}(V)$. If we remove the cyclic condition, there exist examples of non-isomorphic, let alone non-conjugate, subgroups $H$ and $K$ of $\mathrm{GL}(V)$ such that $V\rtimes H\cong V\rtimes K$. Even if we require that non-cyclic subgroups $H$ and $K$ of $\mathrm{GL}(V)$ be abelian, we may still have $V\rtimes H\cong V\rtimes K$ with $H$ and $K$ non-conjugate in $\mathrm{GL}(V)$, but in this case, $H$ and $K$ must at least be isomorphic. If we replace $V$ by a free module $U$ over ${\mathbf Z}/p^m{\mathbf Z}$ of finite rank, with $m>1$, it may happen that $U\rtimes H\cong U\rtimes K$ for non-conjugate cyclic subgroups of $\mathrm{GL}(U)$. If we completely abandon our requirements on $V$, a sufficient criterion is given for a finite group $G$ to admit non-conjugate cyclic subgroups $H$ and $K$ of $\mathrm{Aut}(G)$ such that $G\rtimes H\cong G\rtimes K$. This criterion is satisfied by many groups.

math.GR↗
arXiv2024

The automorphism group of finite $p$-groups associated to the Macdonald group

Given positive integers $p$ and $m$, where $p$ is assumed to be an odd prime, we determine the automorphism groups of $p$-groups $J$, $H$, and $K$ of orders $p^{7m}$, $p^{6m}$, and $p^{5m}$, and nilpotency classes 5, 4, and 3, respectively, all of which arise naturally from the Macdonald group $\langle x,y\,|\, x^{[x,y]}=x^{1+p^m\ell},\, y^{[y,x]}=y^{1+p^m\ell}\rangle$, where $\ell\in{\mathbf Z}$ is relatively prime to $p$.

math.GR↗
arXiv2024

The automorphism group of finite $2$-groups associated to the Macdonald group

We consider the Macdonald group $\langle x,y\,|\, x^{[x,y]}=x^{1+2^m\ell},\, y^{[y,x]}=y^{1+2^m\ell}\rangle$ and its Sylow 2-subgroup $J=\langle x,y\,|\, x^{[x,y]}=x^{1+2^m\ell},\, y^{[y,x]}=y^{1+2^m\ell}, x^{2^{3m-1}}=y^{2^{3m-1}}=1\rangle$, where $m\geq 1$ and $\ell$ is odd. Then $J$ has order $2^{7m-3}$, and nilpotency class 5 if $m>1$ and 3 if $m=1$. We determine the automorphism group of the 2-groups $J$, $H=J/Z(J)$ and $K=H/Z(H)$, where $|H|=2^{6m-3}$ and $|K|=2^{5m-3}$. Explicit multiplication, power, and commutator formulas for $J$, $H$, and $K$ are given, and used in the calculation of $\mathrm{Aut}(J)$, $\mathrm{Aut}(H)$, and $\mathrm{Aut}(K)$.

math.GR↗
arXiv2024

Structure of the Macdonald groups in one parameter

Consider the Macdonald groups $G(α)=\langle A,B\,|\, A^{[A,B]}=A^α,\, B^{[B,A]}=B^α\rangle$, $α\in{\mathbf Z}$. We fill a gap in Macdonald's proof that $G(α)$ is always nilpotent, and proceed to determine the order, upper and lower central series, nilpotency class, and exponent of $G(α)$.

math.GR↗
arXiv2023

On the normalizer of an iterated wreath product

Given a group $G$ and $n\geq 0$, let $W(G,n)$ be the associated iterated wreath product -- unrestricted when $G$ is infinite -- viewed as a permutation group on $G^n$. We prove that the normalizer of $W(G,n)$ in the symmetric group $S(G^n)$ is equal to $M_n\ltimes W(G,n)$, where $M_n$ is isomorphic to~$\mathrm{Aut}(G)^n$. The action of $\mathrm{Aut}(G)^n$ on $W(G,n)$ is recursively described.

math.GR↗
arXiv2021

Linear diophantine equations in several variables

Let $R$ be a ring and let $(a_1,\dots,a_n)\in R^n$ be a unimodular vector, where $n\geq 2$ and each $a_i$ is in the center of $R$. Consider the linear equation $a_1X_1+\cdots+a_nX_n=0$, with solution set $S$. Then $S=S_1+\cdots+S_n$, where each $S_i$ is naturally derived from $(a_1,\dots,a_n)$, and we give a presentation of $S$ in terms of generators taken from the $S_i$ and appropriate relations. Moreover, under suitable assumptions, we elucidate the structure of each quotient module $S/S_i$. Furthermore, assuming that $R$ is a principal ideal domain, we provide a simple way to construct a basis of $S$ and, as an application, we determine the structure of the quotient module $S/U_i$, where each $U_i$ is a specific module containing $S_i$.

math.RA↗
arXiv2021

Linear systems of diophantine equations

Given free modules $M\subseteq L$ of finite rank $f\geq 1$ over a principal ideal domain $R$, we give a procedure to construct a basis of $L$ from a basis of $M$ assuming the invariant factors or elementary divisors of $L/M$ are known. Given a matrix $A\in M_{m,n}(R)$ of rank $r$, its nullspace~$L$ in $R^n$ is a free $R$-module of rank~$f=n-r$. We construct a free submodule $M$ of $L$ of rank~$f$ naturally associated to $A$ and whose basis is easily computable, we determine the invariant factors of the quotient module $L/M$, and then indicate how to apply the previous procedure to build a basis of $L$ from one of $M$.

math.RA↗
arXiv2021

Closed formulae for certain Fermat-Pell equations

Given positive integers $j,k$, with $j\geq 2$, we show that there are positive integers $d,e$ such that $\sqrt{d}$ has continued fraction expansion $\sqrt{d}=[e,\overline{k,\dots,k,2e}]$, with period $j$, if and only if $k$ is even or $3\nmid j$, in which case we give closed formulae to find all such $d,e$ as well as the smallest solution in positive integers to the Fermat-Pell equation $X^2-dY^2=(-1)^j$.

math.NT↗
arXiv2021

On the degree of repeated radical extensions

We answer a question posed by Mordell in 1953, in the case repeated radical extensions, and find necessary and sufficient conditions for $[F[\sqrt[m_1]{N_1},\dots,\sqrt[m_\ell]{N_\ell}]:F]=m_1\cdots m_\ell$, where $F$ is an arbitrary field of characteristic not dividing any $m_i$.

math.NT↗