arXiv subjects

Timothy Kohl

Publications and source records attributed to Timothy Kohl.

10 recordsLinked to original sources

arXiv2020

Hopf Forms and Hopf-Galois Theory

Let $K$ be a finite field extension of $\Q$ and let $N$ be a finite group with automorphism group $F=\Aut(N)$. R. Haggenm\"{u}ller and B. Pareigis have shown that there is a bijection \[\Theta: {\mathcal Gal}(K,F)\rightarrow {\mathcal Hopf}(K[N])\] from the collection of $F$-Galois extensions of $K$ to the collection of Hopf forms of the group ring $K[N]$. For $N=C_n$, $n\ge 1$, $C_p^m$, $p$ prime, $m\ge 1$, and $N=D_3,D_4,Q_8$, we show that $\Q[N]$ admits an absolutely semisimple Hopf form $H$ and find $L$ for which $\Theta (L)=H$. Moreover, if $H$ is the Hopf algebra given by a Hopf-Galois structure on a Galois extension $E/K$, we show how to construct the preimage of $H$ under $\Theta$ assuming certain conditions.

math.RA↗

arXiv2020

Mutually Normalizing Regular Permutation Groups and Zappa-Szep Extensions of the Holomorph

For a group $G$, embedded in its group of permutations $B=Perm(G)$ via the left regular representation $\lambda:G\rightarrow B$, the normalizer of $\lambda(G)$ in $B$ is $\operatorname{Hol}(G)$, the holomorph of $G$. The set $\mathcal{H}(G)$ of those regular $N\leq \operatorname{Hol}(G)$ such that $N\cong G$ and $\operatorname{Norm}_B(N)=\operatorname{Hol}(G)$ is keyed to the structure of the so-called multiple holomorph of $G$, $N\!Hol(G)=\operatorname{Norm}_B(\operatorname{Hol}(G))$, in that $\mathcal{H}(G)$ is the set of conjugates of $\lambda(G)$ by $N\!Hol(G)$. We wish to generalize this by considering a certain set $\mathcal{Q}(G)$ consisting of regular subgroups $M\leq \operatorname{Hol}(G)$, where $M\cong G$, that contains $\mathcal{H}(G)$ with the property that its members mutually normalize each other. This set will generally give rise to a group $Q\!\operatorname{Hol}(G)$ which we will call the quasi-holomorph of $G$, where the orbit of $\lambda(G)$ under $Q\!\operatorname{Hol}(G)$ is $\mathcal{Q}(G)$. The multiple holomorph is a group extension of $\operatorname{Hol}(G)$ and the quasi-holomorph will contain $N\!Hol(G)$, but, when larger than $N\!Hol(G)$, is frequently a Zappa-Sz\'ep product with the holomorph.

math.GR↗

arXiv2019

Enumerating Dihedral Hopf-Galois Structures Acting on Dihedral Extensions

The work of Greither and Pareigis details the enumeration of the Hopf-Galois structures (if any) on a given separable field extension. For an extension $L/K$ which is classically Galois with $G=Gal(L/K)$ the Hopf algebras in question are of the form $(L[N])^{G}$ where $N\leq B=Perm(G)$ is a regular subgroup that is normalized by the left regular representation $\lambda(G)\leq B$. We consider the case where both $G$ and $N$ are isomorphic to a dihedral group $D_n$ for any $n\geq 3$. Using the normal block systems inherent to the left regular representation of each $D_n$,(and every other regular permutation group isomorphic to $D_n$) we explicitly enumerate all possible such $N$ which arise.

math.GR↗

arXiv2018

Characteristic Subgroup Lattices and Hopf-Galois Structures

The Hopf-Galois structures on normal extensions $K/k$ with $G=Gal(K/k)$ are in one-to-one correspondence with the set of regular subgroups $N\leq B=Perm(G)$ that are normalized by the left regular representation $\lambda(G)\leq B$. Each such $N$ corresponds to a Hopf algebra $H_N=(K[N])^G$ that acts on $K/k$. Such regular subgroups $N$ need not be isomorphic to $G$ but must have the same order. One can subdivide the totality of all such $N$ into collections $R(G,[M])$ which is the set of those regular $N$ normalized by $\lambda(G)$ and isomorphic to a given abstract group $M$ where $|M|=|G|$. There arises an injective correspondence between the characteristic subgroups of a given $N$ an d the set of subgroups of $G$ stemming from the Galois correspondence between sub-Hopf algebras of $H_N$ and intermediate fields $k\subseteq F\subseteq K$. We utilize this correspondence to show that for certain pairings $(G,[M])$, the collection $R(G,[M])$ must be empty.

math.GR↗

arXiv2017

Isomorphism problems for Hopf-Galois structures on separable field extensions

Let $ L/K $ be a finite separable extension of fields whose Galois closure $ E/K $ has group $ G $. Greither and Pareigis have used Galois descent to show that a Hopf algebra giving a Hopf-Galois structure on $ L/K $ has the form $ E[N]^{G} $ for some group $ N $ such that $ |N|=[L:K] $. We formulate criteria for two such Hopf algebras to be isomorphic as Hopf algebras, and provide a variety of examples. In the case that the Hopf algebras in question are commutative, we also determine criteria for them to be isomorphic as $ K $-algebras. By applying our results, we complete a detailed analysis of the distinct Hopf algebras and $ K $-algebras that appear in the classification of Hopf-Galois structures on a cyclic extension of degree $ p^{n} $, for $ p $ an odd prime number.

math.NT↗

arXiv2017

The Structure of Hopf Algebras Acting on Dihedral Extensions

We discuss isomorphism questions concerning the Hopf algebras that yield Hopf-Galois structures for a fixed separable field extension $L/K$. We study in detail the case where $L/K$ is Galois with dihedral group $D_p$, $p\ge 3$ prime and give explicit descriptions of the Hopf algebras which act on $L/K$. We also determine when two such Hopf algebras are isomorphic, either as Hopf algebras or as algebras. For the case $p=3$ and a chosen $L/K$, we give the Wedderburn-Artin decompositions of the Hopf algebras.

math.NT↗

arXiv2017

Normality and Short Exact Sequences of Hopf-Galois Structures

Every Hopf-Galois structure on a finite Galois extension $K/k$ where $G=Gal(K/k)$ corresponds uniquely to a regular subgroup $N\leq B=\operatorname{Perm}(G)$, normalized by $\lambda(G)\leq B$, in accordance with a theorem of Greither and Pareigis. The resulting Hopf algebra which acts on $K/k$ is $H_N=(K[N])^{\lambda(G)}$. For a given such $N$ we consider the Hopf-Galois structure arising from a subgroup $P\triangleleft N$ that is also normalized by $\lambda(G)$. This subgroup gives rise to a Hopf sub-algebra $H_P\subseteq H_N$ with fixed field $F=K^{H_P}$. By the work of Chase and Sweedler, this yields a Hopf-Galois structure on the extension $K/F$ where the action arises by base changing $H_P$ to $F\otimes_k H_P$ which is an $F$-Hopf algebra. We examine this analogy with classical Galois theory, and also examine how the Hopf-Galois structure on $K/F$ relates to that on $K/k$. We will also pay particular attention to how the Greither-Pareigis enumeration/construction of those $H_P$ acting on $K/F$ relates to that of the $H_N$ which act on $K/k$. In the process we also examine short exact sequences of the Hopf algebras which act, whose exactness is directly tied to the descent theoretic description of these algebras.

math.NT↗

arXiv2016

A Class of Profinite Hopf-Galois Extensions Over Q

For $p$ a prime and $a\in\mathbb{Q}$, where $a$ is not a $p^n$-th power of any rational number, the extension $\mathbb{Q}(w_n)/\mathbb{Q}$ where $w_n=\root p^n \of a$ is separable but non-normal. The Hopf-Galois theory for separable extensions was determined by Greither and Pareigis, and the specific classification for radical extensions such as these by the author. In this work we extend this theory to a certain class of profinite extensions, namely those formed from the union of these $\mathbb{Q}(w_n)$. We construct a 'profinite' Hopf algebra which acts, and show that it satisfies a generalization of a result due to Haggenmuller and Pareigis on the structure of Hopf algebra forms of group algebras.

math.RA↗

arXiv2014

Hopf-Galois Structures Arising From Groups with Unique Subgroup of Order p

For $\Gamma$ a group of order $mp$ for $p$ prime where $gcd(p,m)=1$, we consider those regular subgroups $N\leq Perm(\Gamma)$ normalized by $\lambda(\Gamma)$, the left regular representation of $\Gamma$. These subgroups are in one-to-one correspondence with the Hopf-Galois structures on separable field extensions $L/K$ with $\Gamma=Gal(L/K)$. This is a follow up to the author's earlier work where, by assuming $p>m$, one has that all such $N$ lie within the normalizer of the $p$-Sylow subgroup of $\lambda(\Gamma)$. Here we show that one only need assume that all groups of a given order $mp$ have a unique $p$-Sylow subgroup, and that $p$ not be a divisor of the automorphism groups of any group of order $m$. As such, we extend the applicability of the program for computing these regular subgroups $N$ and concordantly the corresponding Hopf-Galois structures on separable extensions of degree $mp$.

math.GR↗

arXiv2002

Cyclotomic Swan subgroups and primitive roots

Let $K_{m}=\Bbb{Q}(ζ_{m})$ where $ζ_{m}$ is a primitive $m$th root of unity. Let $p>2$ be prime and let $C_{p}$ denote the group of order $p.$ The ring of algebraic integers of $K_{m}$ is $\Cal{O}_{m}=\Bbb{Z}[ζ_{m}].$ Let $Λ_{m,p}$ denote the order $\Cal{O}_{m}[C_{p}]$ in the algebra $K_{m}[C_{p}].$ Consider the kernel group $D(Λ_{m,p})$ and the Swan subgroup $T(Λ_{m,p}).$ If $(p,m)=1$ these two subgroups of the class group coincide. Restricting to when there is a rational prime $p$ that is prime in $\Cal{O}_{m}$ requires $m=4$ or $q^{n}$ where $q>2$ is prime. For each such $m$, $3 \leq m \leq 100,$ we give such a prime, and show that one may compute $T(Λ_{m,p})$ as a quotient of the group of units of a finite field. When $h_{mp}^{+}=1$ we give exact values for $|T(Λ_{m,p})|$, and for other cases we provide an upper bound. We explore the Galois module theoretic implications of these results.

math.NT↗