Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,182 questions
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Is every finite $d$-dimensional matrix group generated by $d$ elements?
The question is in the title. If $\Gamma\subset\mathrm{GL}(\Bbb R^d)$ is a finite matrix group, can it be generated by (at most) $d$ elements?
I suspect that this hope is too naive, but I have no ...
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Prehistory of Gromov-hyperbolic spaces/groups
When speaking about hyperbolic groups/spaces, one usually refers to Gromov's monograph Hyperbolic groups for their introduction. However, coarse notions of hyperbolicity can be found in some of his ...
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Laws characterizing the trivial group
What are laws characterizing the trivial group? I mean all group words $w$ such that if the identity $w=1$ holds in a group $G$, then $G=1$.
For example, it can be easily verified that if a group ...
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Does a compact negatively curved manfiold of dimension 4 admit a cover of finite degree?
A $3$-dimensional compact manifold of negative sectional curvature admits (by geometrisation?) a metric of curvature $-1$, and so its fundamental group has subgroups of finite index. I wonder if an ...
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AC in group isomorphism between R and R^2
Using the axiom of choice, one can show that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as additive groups. In particular, they are both vector spaces over $\mathbb{Q}$ and AC gives bases of ...
- 8,501
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Generators for the first cohomology of free groups
Let $F = \langle x_1, \dots, x_n \rangle$ be the free group on $n$ generators and $R = \mathbb Z$. The Fox derivatives $\frac{\partial}{\partial x_i} \colon F \to R[F]$ are the unique derivations ...
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Cartography of the duals of GL, PGL, SL, etc
A short version of this question could be
What are the duals of $PGL(2,\mathbf{Q}_p)$, $PGL(2,\mathbf{R})$ and $PGL(2,\mathbf{C})$?
I should obviously add some precisions.
there are different ...
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Steinberg representation for sporadic simple groups?
The Steinberg representation is a remarkable irreducible representation of a reductive algebraic group over a finite field or local field, or a group with a BN-pair. It is analogous to the 1-...
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The Möbius number of the nonabelian finite simple groups
Let $L$ be a finite lattice with minimum $\hat{0}$ and maximum $\hat{1}$. The Möbius function $\mu$ for $L$ is defined recursively by: for $\forall a,b \in L$ with $a<b$, $\mu(b,b) = 1$ and $\mu(...
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Groups with triple system of self-normalizing subgroups
Does there exist a group $G$ (finite or infinite) with three subgroups $A, B, C \leq G$ satisfying the following three conditions?
$A = N_G(A)$, $B = N_G(B)$, $C = N_G(C)$;
$AB = BC = CA = G$;
$A \...
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Group with fewer than $p^2$ Sylow p-subgroups
(This question is originally from Math.SE where it was suggested that I ask the question here)
Let $G$ be a finite group with fewer than $p^2$ Sylow $p$-subgroups, and let $p^n$ be the power of $p$ ...
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Example of reflective subcategory of (Groups) whose reflector doesn't preserve finite products
A subcategory $D$ of a category $C$ is called reflective, if the embedding $D \hookrightarrow C$ has a left adjoint $L:C \to D$. The left adjoint $L$ is called the reflector. If the category $C$ is ...
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Does such an infinite index subgroup exist?
Notation: If $G$ is a countable group and $H$ is a subgroup, for $g\in G$, let $|\mathcal{O}_{gH}|$ be the size of the $H$-orbit of $gH$ in the $H$-set $G/H$.
Does there exist a countable group $G$ ...
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Does every automorphism of G come from an inner automorphism of S_G?
I feel sort of silly asking this question. Unless I'm very much mistaken the paper I'm reading assumes the following statement:
Let $G$ be a finite group. We may embed it via the Cayley embedding ...
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Embedding rank of finite groups and quotients
Let $G$ be a finite group, and $n$ a positive integer. It is not hard to check that the following are equivalent:
For every $g\in G\setminus\{1\}$ there is a subgroup $H\leq G$ with $|G/H|\leq n$ ...
- 56.9k
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What is a "general" relation algebra?
I'm trying to understand why (or if) the axioms of relation algebras are "the right ones." For example, we can back up the idea that the group axioms exactly capture the notion of "...
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Is $\text{PSL}_2(\mathbb{F}_{p^m})$ known to be a Galois group over $\mathbb{Q}$ for $m>1$?
Let $\mathbb{F}$ be a finite field of characteristic $p$, is it known that $\text{PSL}_2(\mathbb{F})$ can be realized as a Galois extension of $\mathbb{Q}$ for any/all cases when $\mathbb{F}$ is not $\...
user130124
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Weil's book L'intégration dans les groupes topologiques et ses applications
The book L'intégration dans les groupes topologiques et ses applications published by André Weil in 1940 is regarded as one of the classical references for harmonic analysis on topological groups.
...
user92170
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Which groups are LERF?
A finitely generated group $G$ is called LERF if every finitely generated $H \leq G$ is closed in the profinite topology on $G$ (equivalently, there is a family of finite index subgroups of $G$ ...
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Do subgroups have "two sided bases"?
Let $H\leq G$ be an inclusion of finite groups. Define a map $E\colon \mathbb{C}[G]\to \mathbb{C}[H]$ to be the $\mathbb{C}$-linear extension of
$$
E(g)=\begin{cases}
g &\text{if } g\in H\\\
0 &...
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Bass's paper "Symplectic groups and modules", used in proof of the congruence subgroup property for Sp
Let $R$ be the ring of integers in a number field. While studying the congruence subgroup property for $\text{Sp}_{2g}(R)$ in
Bass, H.; Milnor, J.; Serre, J.-P.
Solution of the congruence subgroup ...
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Which finite groups have low-degree essential cohomology?
Let $G$ be a finite group, $A$ some coefficients (e.g. $A = \mathbb{F}_2$ or $\mathbb{Z}$), and write $\mathrm{H}^\bullet_{\mathrm{gp}}(G; A)$ for the (ordinary) group cohomology of $G$ with ...
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Does this group construction preserve finite presentability?
Suppose $G$ is a group. Consider the set $G^G$ of all functions $G \to G$, which forms a group under elementwise multiplication. Now, for all $g \in G$ let’s define $c_g \in G^G$ as the constant ...
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How nearly abelian are nilpotent groups?
It is not uncommon to read that "nilpotent groups are 'close to abelian'."1,2
Can this sentiment be made precise
in the sense of the
Turán and Erdős definition of "the probability that two elements of ...
- 151k
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Hilbert's 10th problem and nilpotent groups
I am asking this question on behalf of a colleague of mine who does not have an MO account. Nevertheless I am also interested in the answer.
The question concerns relationships between Hilbert's ...
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a normal form for matrices over Z[x]/(x^2-1) ?
We are discussing, offline, modules over the $\mathbb{Z}$-group ring of the cyclic group of order 2, which is probably better known as the quotient ring $R=\mathbb{Z}[t]/(t^2-1)$.
Is there any way to ...
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Is there a characterization of free groups in terms of the unitary dual?
If $G$ is a countable discrete group, I'm curious if it is possible to decide whether $G$ is a free group only by looking at properties of $Rep(G)$, the collection of (equivalence classes of) strongly ...
- 7,067
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Decidability of word problem for group admitting certain action
Let $G$ be a group acting highly transitively (and faithfully) on a set $S$. Suppose that $G$ is finitely presented, and that every stabilizer in $G$ of a finite subset of $S$ is finitely generated. I ...
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Every subgroup is isomorphic to a normal subgroup
Let $G$ be a group such that, for every subgroup $H$ of $G$, there exists a normal subgroup $K$ of $G$, such that $H$ is isomorphic to $K$. Under such conditions, can we determine the structure of $G$ ...
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For what sets $X$ do there exist a pair of functions from $X$ to $X$ with the identity being the only function that commutes with both?
It is not too difficult to show that if $X$ is an infinite set, then there exists a two-element subset of the group $\operatorname{Sym}(X)$ with trivial centralizer iff $\lvert X\rvert \leq \lvert\...
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Is every finite-order unimodular matrix conjugate to a $0,1,-1$ matrix?
Problem. Given a matrix $A\in\mathrm{GL}(n,\mathbb{Z})$ such that $A^k=1$ for some $k\geq 1$, is there a matrix $g\in\mathrm{GL}(n,\mathbb{Z})$ such that $gAg^{-1}$ has only $0$, $1$, and $-1$ as ...
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Isometries of some simple Cayley graphs
Consider a Cayley graph of a group $G$ with respect to a symmetric finite generating set $S$.
There are some obvious candidates to isometries of this graph - for example, translation by elements of $G$...
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Generalization of a theorem of Øystein Ore in group theory
Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive.
Proof: see below.
Let $(H \subset G)$ be an inclusion of finite groups and $\...
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Simple groups with the same cardinality as PSL_2(Z/p)
In an undergrad honors algebra course it's sometimes shown that when $p$ is prime and $>3$ then
$PSL_2(Z/p)$ is simple of of order $p(p-1)(p+1)/2$. But that this is the "only" simple group
having ...
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Does every f.g. group have a minimal presentation?
Call a group presentation $\langle X \,\|\,R \rangle$ minimal if no relator from $R$ is a consequence of the remaining relators, i.e., no $r \in R$ belongs to the normal closure of $R\setminus \{r\}$ ...
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The finiteness criterium $F$ under quasi-isometry
A group $G$ is defined to have $F$ if there exists a finite $K(G,1)$.
This property is clearly not invariant under quasi-isometry as one can see from the trivial group and $\mathbb{Z}_2$.
My question:...
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Finite index subgroups of free groups and torsion-free amenable quotients of free groups
Let $F$ be a finitely generated free group and let $A$ be a finite index subgroup of $F$.
Does there exist a subgroup $B\subset A$ such that $F/B$ is (elementary) amenable and torsion-free?
A group $...
- 2,797
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Is the following map from Z(G) x H^3(G, C*) --> H^2(G, C*) ever nontrivial?
Suppose that G is a finite group, then we have the following map f which takes an element z in the center of G and a 3-cohomology class w and returns a 2-cohomology class f(z,w) (for concreteness let'...
- 28.1k
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Number of finite groups: is $\operatorname{gnu}(4n) \geq 2 \operatorname{gnu}(n)$?
Let $\operatorname{gnu}(n)$ be the number of finite groups of order $n$.
Question: Is it true that $\operatorname{gnu}(4n) \geq 2 \operatorname{gnu}(n)$ for all $n \geq 1$?
Surely this must be true, ...
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Mathematical explanation of orbital shell sizes: why is it sufficient to consider single-electron wave functions?
Motivation
The question "Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?" asks for an explanation of the sequence 2, 8, 8, ...
- 2,549
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668
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On the sum of the subgroup orders of a finite group
Let $G$ be a finite group. Consider the function providing the sum of the subgroups orders
$$\sigma(G) = \sum_{H \le G} |H|.$$
Note that if $C_n$ is cyclic of order $n$ then $\sigma(C_n) = \sigma(n)$, ...
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Dimensional gap of group representations
The problem is inspired by eigenvalue bounds of random Cayley graphs on $SL_2(q)$.
Definition. An infinite series of finite groups $S$ is α-rich if the dimension of the smallest nontrivial ...
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Difference between the completed group algebra and the profinite completion of a group ring
Let $G$ be a reasonably nice group, say residually finite if need be.
We may consider the group algebra $\mathbb{Z}[G]$.
Let $\widehat{\mathbb{Z}[G]} := \varprojlim_I\mathbb{Z}[G]/I$ be the ...
- 7,527
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Some questions about the Malcev completion
Let $G$ be an abstract group. The Malcev completion $\widehat{G}$ of $G$ (over $\mathbb{Q}$) is the set of group-like elements in the complete Hopf algebra $\widehat{\mathbb{Q}[G]} = \lim_n \mathbb{Q}...
- 4,484
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What are smallest finite images of triangle groups?
Given integers $a, b, c$ all at least 2, I would like to identify the smallest group with an element $x$ of order $a$, element $y$ of order $b$ such that the product $yx$ has order $c$. For example, ...
- 1,021
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Gromov's Hyperbolicity and Positive Cheeger Constant in Planar Graphs
It is known that for metric graphs the concepts of Gromov's hyperbolicity and strictly positive Cheeger constant are related. Let us first recall the definition of the Cheeger constant. Let $G$ be a ...
- 3,626
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Does the Poincare series of a Coxeter group always describe a "flag variety"?
Let $W$ be a Coxeter group and let $P_W(q) = \sum_{w \in W} q^{\ell(w)}$ be its Poincare series. When $W$ is the Weyl group of a simple algebraic group $G$ (hence $W$ is finite), $P_W(q)$ is the ...
- 118k
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Topologies on an infinite symmetric group
Let $X$ be an infinite set, and let $G$ be the symmetric group on $X$. I want to understand $G$ by putting a topology on it, without imposing any more structure on $X$. What 'interesting' ...
- 4,728
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Minimal length presentations of cyclic groups
By the length of a finite presentation I mean the sum of the lengths of the relators. I am interested in knowing what the minimal length of a presentation of $\mathbb{Z}/n\mathbb{Z}$. I'm even more ...
- 5,349
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Is there a name of semidirect product of a group with its automorphism group?
Consider the construction $G \rtimes \text{Aut}(G)$. Here $
G$ is a group, $\text{Aut}(G)$ is the automorphism group and the semidirect product is over the most obvious action.
1) Is there any name ...
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