Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
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Reachability threshold in rulial space
I have been getting interested in so-called "Theories of Everything" and have been reading Stephen Wolfram's work. This problem is inspired by the idea of the ruliad.
Let $\mathcal{T}_{s,k}$ ...
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Centralizer of PSL in PGL and of SL in GL: reference request
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\PSL{PSL}$Consider the general linear group $\GL(n,q)$ over a finite field with $q$ elements and ...
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Is the number of varieties of groups still unknown?
A variety of groups is a class of groups satisfying a specified set of equations. Equivalently, it is a class of groups that is closed under homomorphic images, subgroups, and direct products. A ...
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Loewy length of $F_pQ$ for $Q$ Sylow p-group in $S_n$
I am trying to upper bound the Davenport constant for $S_n$ in my research work and in order to do I have the following hint given by @DenisT (thanks a lot!) in another post (Upper bound for Davenport ...
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In dimension $n=5$, does a subgroup of $O(n)$ satisfying these properties exist?
I asked a question where @YCor provided a construction that seems to enable a group construction satisfying some properties when $n\ne 5$. However, in the case $n=5$, I am starting to think no such ...
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Characterization of Vilenkin group
It is shown in [1, Section 1] by C.W. Onneweer that every infinite compact, metrizable, zero-dimensional commutative group is a Vilenkin group. My question is does this implication also hold if we ...
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Finitely dominated universal spaces for the family of solvable subgroups
$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Sz{Sz}$In short, I am interested in the question which finite groups $G$ admit a finitely dominated universal space with respect to the family of ...
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Non-commuting elements of finite orders in a reductive group over a p-adic field
Let $k$ be a $p$-adic field and $G$ be a connected non-abelian reductive algebraic group over $k$. I am asking for a proof of the following lemma:
Lemma. Assuming that $p$ is "good" for $G$,...
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Group generated by "adjacent" permutations of graph
Let $G=(V,E)$ be any graph, i.e. $E$ is simply a binary relation over $V$. We say a permutation $\sigma\in \text{Bij}(V)$ of the vertices of $G$ is adjacent if, for all $v\in V$, $(v,\sigma(v))\in E$. ...
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Groups without infinitely divisible elements [closed]
Let $(G,\cdot)$ be a group. Call an element $g \in G$ infinitely divisible if there exist infinitely many positive integers $N$ such that $h^N = g$ for some $h \in G$.
Question: is there an easy way ...
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Function related to length of group presentations: is it computable?
(This question comes from a friend who works in sofic group theory.)
Consider the function $f: \mathbb{N} \to \mathbb{N}$, defined, for any $n \in \mathbb{N}$, by putting $f(n)$ to be the largest ...
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Do linear groups over a commutative ring satisfy the Tits alternative?
A group $G$ is said to satisfy the Tits alternative if any finitely generated subgroup of $G$ is either virtually solvable or contains a nonabelian free subgroup. Tits proved this for linear groups ...
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Does every cancellative duo semigroup embed into a group?
Prompted by the comments to a recent answer by YCor to a related question (here), I'd like to ask the following:
Q. Does every cancellative duo semigroup embed into a group?
A (multiplicatively ...
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Continuous functions on HLS groupoids
I am reading a paper about property (T) for groupoids: Topological property (T) for groupoids. In section 4.4 they discuss the HLS groupoids which I describe define here.
Let $\Gamma$ be a discrete ...
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The base group of a wreath product of an abelian group by $ {\mathbb{Z}}$ is a characterstic subgroup
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can direct me to some relevant results.
Let $A$ be a finitely generated abelian group,...
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