Related papers: Minimal cover groups

This is the second installment of an exposition of an ACL2 formalization of finite group theory. The first, which was presented at the 2022 ACL2 workshop, covered groups and subgroups, cosets, normal subgroups, and quotient groups,…

For a semigroup $S$, the covering number of $S$ with respect to semigroups, $\sigma_s(S)$, is the minimum number of proper subsemigroups of $S$ whose union is $S$. This article investigates covering numbers of semigroups and analogously…

Let ${\frak F}$ be a class of group and $G$ a finite group. Then a set $\Sigma $ of subgroups of $G$ is called a \emph{$G$-covering subgroup system} for the class ${\frak F}$ if $G\in {\frak F}$ whenever $\Sigma \subseteq {\frak F}$. We…

For a prime power $q$ and a positive integer $n$, we say a subspace $U$ of ${\mathbb{F}_q^n}$ is {\em cyclically covering} if the union of the cyclic shifts of $U$ is equal to $\mathbb{F}_q^n$. We investigate the problem of determining the…

A set of proper subgroups is a covering for a group if its union is the whole group. The minimal number of subgroups needed to cover $G$ is called the covering number of $G$, denoted by $\sigma(G)$. Determining $\sigma(G)$ is an open…

For a group $G$ and a natural number $m$, a subset $A$ of $G$ is called $m$-thin if, for each finite subset $F$ of $G$, there exists a finite subset $K$ of $G$ such that $|Fg\cap A|\leqslant m$ for every $g\in G\setminus K$. We show that…

We introduce the notion of the depth of a finite group $G$, defined as the minimal length of an unrefinable chain of subgroups from $G$ to the trivial subgroup. In this paper we investigate the depth of (non-abelian) finite simple groups.…

Previous formulations of group theory in ACL2 and Nqthm, based on either "encapsulate" or "defn-sk", have been limited by their failure to provide a path to proof by induction on the order of a group, which is required for most interesting…

We know that any finite abelian group $G$ appears as a subgroup of infinitely many multiplicative groups $\mathbb{Z}_n^\times$ (the abelian groups of size $\phi(n)$ that are the multiplicative groups of units in the rings…

A group is small if it has countably many complete $n$-types over the empty set for each natural number n. More generally, a group $G$ is weakly small if it has countably many complete 1-types over every finite subset of G. We show here…

We prove:(1) the existence, for every integer n > 3, of a noncompact smooth n-dimensional topological manifold whose diffeomorphism group contains an isomorphic copy of every finitely presented group; (2) a finiteness theorem on finite…

For a group $G$, a {\it normalizer covering} of $G$ is a finite set of proper normalizers of some subgroups of $G$ whose union is $G$. We study $p$-groups ($p$ a prime) without a normalizer covering. As an application, we determine some…

A fundamental notion in group theory, which originates in an article of Ulam and von Neumann from $1947$ is uniform simplicity. A group $G$ is said to be $n$-uniformly simple for $n \in \mathbf{N}$ if for every $f,g\in G\setminus \{id\}$,…

A Hausdorff topological group G is minimal if every continuous isomorphism f : G --> H between G and a Hausdorff topological group H is open. Clearly, every compact Hausdorff group is minimal. It is well known that every infinite compact…

Let $T$ be a theory which is t-minimal, meaning that with respect to some definable topology, a unary definable set $D \subseteq M$ has non-empty interior iff it is infinite. If $K$ is a definable field in $T$, then $K$ is finite or "large"…

Given a finite non-cyclic group $G$, call $\sigma(G)$ the smallest number of proper subgroups of $G$ needed to cover $G$. Lucchini and Detomi conjectured that if a nonabelian group $G$ is such that $\sigma(G) < \sigma(G/N)$ for every…

A finite $p$-group $G$ is called \textit{powerful} if either $p$ is odd and $[G,G]\subseteq G^p$ or $p=2$ and $[G,G]\subseteq G^4$. A {\em{cover}} for a group is a collection of subgroups whose union is equal to the entire group. We will…

A profinite group is called small if it has only finitely many open subgroups of index n for each positive integer n. We show that every Frattini cover of a small profinite group is small. A profinite group is called strongly complete if…

We determine the groups of minimal order in which all groups of order n can embedded for 1 < n < 16. We further determine the order of a minimal group in which all groups or order n or less can be embedded, also for 1 < n < 16.

A Hausdorff topological group G is minimal if every continuous isomorphism f: G --> H between G and a Hausdorff topological group H is open. Significantly strengthening a 1981 result of Stoyanov, we prove the following theorem: For every…