Let $F_n$ be a free group of rank $n\geq3$, freely generated by $x_1,\ldots,x_n$. For $1\leq j<i\leq n$, let $d_{i,j}$ denote the Nielsen automorphism of $F_n$ defined by $d_{i,j}(x_i)=x_ix_j$ and $d_{i,j}(x_k)=x_k$ for $k\neq i$, and let $D_n=\langle d_{i,j}\mid 1\leq j<i\leq n\rangle$. We determine the lower central series of $D_n$. We first prove that, for $1\leq r<i\leq n$, $d_{i,r}\in \gamma_{i-r}(D_n)\setminus\gamma_{i-r+1}(D_n)$. For each $i=2,\ldots,n$, this calculation leads to a filtration $\{W_{i,c}\}_{c\geq1}$ of $U_i=\langle d_{i,1},\ldots,d_{i,i-1}\rangle$. For every $c\geq1$, we obtain an explicit iterated semidirect-product decomposition of $\gamma_c(D_n)$ in terms of the subgroups $W_{i,c}$, and prove that $U_i\cap\gamma_c(D_n)=W_{i,c}$ for $i=2,\ldots,n$. The construction also gives an explicit basis for each quotient $\gamma_c(D_n)/\gamma_{c+1}(D_n)$ in terms of basic commutators and determines the exact lower-central depth of every such commutator. Consequently, $D_n$ is a Magnus group.
Let $F_n$ be the free group of rank $n\geq2$ with basis $x_1,\ldots,x_n$. For $1\leq j<i\leq n$, let $d_{i,j}$ and $e_{i,j}$ be the automorphisms of $F_n$ defined by $d_{i,j}(x_i)=x_ix_j$ and $e_{i,j}(x_i)=x_jx_i$, respectively, and fixing the remaining free generators. Write $D_n=\langle d_{i,j}\mid 1\leq j<i\leq n\rangle$ and $A_n^+=\langle d_{i,j},e_{i,j}\mid 1\leq j<i\leq n\rangle$. We prove that $D_n$ admits a presentation on the generators $d_{r+1,r}$, $1\leq r\leq n-1$, with three families of relations given by commutators of weights two, three, and four, respectively. We extend this to a presentation of $A_n^+$ on the generators $d_{r+1,r}$ and $e_{r+1,r}$, $1\leq r\leq n-1$. These generating sets have minimum cardinality. Moreover, the presentation of $A_n^+$ may be chosen so that every defining relator is a single commutator.
For $n\geq 4$, let $T_n$ be the rank-3 matroid on $\mathbb{Z}/n\mathbb{Z}$ whose bases are the three-element non-zero-sum subsets. Let $X_1(n)^\circ$ denote the open subscheme of the modular curve $X_1(n)$ obtained by removing the cusps corresponding to reducible N\'eron polygons. For $n \geq 10$, we give a purely algebraic and incidence-theoretic proof that, for every field $k$ with $\mathrm{char}(k)$ not dividing $n$, there is a natural bijection between $X_1(n)^\circ(k)$ and rescaling classes of $k$-realizations of $T_n$. For $k = \mathbb{C}$, this recovers a theorem of Borisov and Roulleau. We then upgrade the field-valued correspondence to an isomorphism of schemes over $\mathbb{Z}[1/n]$. The main new ingredient is a deformation-theoretic argument which allows us to verify the isomorphism on points valued in Artinian local rings. As consequences, the modular curve $X_1(n)^\circ$ acquires a natural model over $\mathbb{Z}[1/n]$ as a matroid realization space, and, for primes $p \geq 11$, the non-representability of $T_p$ over $\mathbb{Q}$ is equivalent to the prime-order case of Mazur's celebrated theorem on rational torsion points of elliptic curves. In an appendix, we explain how to upgrade the realization space of a matroid from an affine scheme over $\mathbb{Z}$ to an affine band scheme (in the sense of Baker-Jin-Lorscheid) over $\mathbb{F}_1^{\pm}$.
Let $E_n$ denote the number of alternating permutations of $\{1,\dots,n\}$, equivalently characterized by $\sum_{n\ge0}E_nz^n/n!=\sec z+\tan z$. For every $q\ge1$, the sequence $(E_n\bmod q)_{n\ge0}$ is eventually periodic; let $d(q)$ and $s(q)$ denote its minimal eventual period and preperiod. For every odd prime $p$, Knuth and Buckholtz proved $d(p)=\operatorname{lcm}(p-1,4)$ together with \[ d(p^r)\mid p^{r-1}d(p), \qquad s(p^r)\le r, \] and Ramassamy conjectured that both bounds are attained for every $r\ge1$. In this paper, we introduce an algebraic frequency expansion for the Euler zig-zag numbers using Hurwitz series over the coefficient ring $S_r=(\mathbb Z/p^r\mathbb Z)[x]/(x^2+1)$. More precisely, the corresponding Hurwitz series is represented as a finite combination of formal exponential modes, in a manner reminiscent of Fourier analysis.Using this expansion, we prove \[ d(p^r)=p^{r-1}d(p) \qquad \text{for every odd prime $p$ and every $r\ge1$}, \] thereby establishing Ramassamy's period conjecture. We also disprove the preperiod conjecture by proving \[ s(5^5)=4<5. \] Finally, we prove that $5^5$ is the smallest odd prime power for which $s(p^r)\ne r$, and based on our findings we further conjecture \[ s(p^r)\ge r-2 \] for every odd prime $p$ and every $r\ge2$.
A subspace of $\mathbb{F}_q^n$ is called cyclically covering if the union of $\sigma^i(U)$ can cover the whole space $\mathbb{F}_q^n$, where $\sigma$ is the cyclic shift, $0 \leqslant i \leqslant n-1$. Let $h_q(n)$ be the largest possible co-dimension of a cyclically covering subspace of $\mathbb{F}_q^n$. We show that $h_2(2p) = 2$ for every prime $p$ such that $2$ is a primitive root modulo $p$. By constacyclic codes, we show that $h_q((q-1)n) = 0$ when $h_q(n) = 0$ and $\gcd(n,q-1) = 1$. We also derive a lower bound on $h_q(n)$ by the concept of support weight distribution, which is important in coding theory. Finally, using irreducible cyclic codes, we present several families of $n$ such that $h_q(n) = 0$.
Let $r>1$ with $\gcd(r,30)=1$. We construct a finite group $G_r=(C_5\times C_3\times C_r)\rtimes W$, where $|W|=32$ and $G_r'\cong C_{60r}$, together with an augmentation-preserving automorphism $\alpha_r\in\operatorname{Aut}(\mathbb{Z}G_r)$ having no Zassenhaus factorization. The image $Y_r=\alpha_r(G_r)$ is a normalized group basis which is not rationally conjugate to $G_r$, although every element of $Y_r$ is individually rationally conjugate to an element of $G_r$. Consequently, (ZC2) and (ZC3) fail for finite cyclic-by-abelian groups, resolving a problem of Margolis and del R\'io. The construction extends Hertweck's example uniformly: both the class-preserving obstruction and the integral gluing are independent of the order of the auxiliary factor. The smallest admissible member of this family has order $3360$ and derived subgroup $C_{420}$.
Let $q$ be a primitive $n$-th root of unity, $n>1$, and let $f$ be a nonzero polynomial such that $n\nmid(j+1)$ for every $j\in\supp(f)$. Set $e=\gcd(n,\{j+1:j\in\supp(f)\})$. We show that $\Oz(B_q(f))\cong\mu_e\times\mu_e$: the defining relations are homogeneous for a $\mathbb{Z}/e\times\mathbb{Z}/e$ grading, the center sits in degree zero, and the ozone group is the character group of that grading. The determination of the ozone group only requires the central elements $u^n$, $v^n$, and $\Omega$. The regular normal elements modulo the center form the same group, generated by $u^{n/e}$ and $v^{n/e}$, so every normal element is central exactly when $e=1$. For $f=t^2$ we recover a computation of Chan, Gaddis, Won and Zhang, and for $e>1$ we obtain an infinite family of Calabi--Yau algebras with nontrivial ozone group.
James G'omez, Helbert Venegas· 0 citations
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