Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
7,942
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Subgroup lattice isomorphic to the power set lattice
If $G$ is a group, we denote by $\text{Sub}(G)$ the lattice of all subgroups of $G$, ordered by $\subseteq$. Given a cardinal $\kappa$, is there a group $G$ with $\text{Sub}(G) \cong {\cal P}(\kappa)$ ...
- 45.5k
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Amenable Thompson-like groups
Question: Do there exist amenable Thompson-like groups?
I realise that my question is vague, but defining and studying groups which look like Thompson's groups $F$, $T$ and $V$ seems to be an ...
- 7,511
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A robust version of Schur's lemma?
Does a robust version of Schur's lemma exist? Specifically, I was wondering about something like this:
Let $B$ be a bounded operator over a vector space $V$, with underlying field $\mathbb{C}$ and ...
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2
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Distance regular Cayley graphs on $Z_2^n$?
Let $Z_2^n$ be group $Z_2 \times Z_2 \times \cdots \times Z_2$ with operation Exclusive-or. I'd like to know if the $Cay(Z_2^n,S)$ for $S \subset Z_2^n \setminus \{0\}$ is distance regular graph or ...
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About the product of finite subsets of a torsion free group
Let $G$ be a torsion free group with identity $e$. For a subset $X$ of $G$, denote by $X^\#$ the set $X\setminus\{e\}$. Let $A$ be a finite subset of $G$ containing $e$. Is there a finite subset $B$ ...
- 2,118
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representing an uncountable free group as a union of an increasing sequence of countable subgroups
Let $(G_\alpha)$ and $(K_\alpha)$ $(\alpha<\aleph_1)$ be strictly increasing chains of countable sets such that if $\alpha$ is a limit, then $G_\alpha=\bigcup_{\beta<\alpha}G_\beta$ and $K_\...
- 1,388
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Example of a ring with non-finitely generated unit group?
The well known Dirichlet's unit theorem states that the unit group of a maximal order in a quadratic number field is finitely generated of rank blah blah blah. I think it's pretty naive to expect a ...
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Character degrees of a finite group?
Let $G$ be a finite group, $R(G)$ be the solvable radical of $G$, and $G/R(G)$ be an almost simple group with socle $S\cong PSL_2(3^p)$, where $p$ is an odd prime. Also let $[G/R(G):S]=p$ and $\theta$ ...
- 841
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Ordered group acting freely on partially ordered set
Let $(G, <)$ be a totally ordered group, and let $<$ be left-invariant. Let $G$ act (freely?) on a partially ordered set $(S, <)$, such that this group action preserves the ordering:
$$ s_1 &...
- 51
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What is the complexity of determining if a knot group is $\mathbb{Z}$?
It is known from the work of Waldhausen that the isomorphism problem for knot groups is decidable. What is then:
The complexity of determining if a knot group is $\mathbb{Z}$? .i.e. same as the ...
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A question about spectral properties of a non-amenable group
Let $G$ be a group generated by $a,b$ (for the sake of simplicity). Consider the element
$$S=a+b+a^{-1}+b^{-1}\in{\mathbb C}[G],$$
which may also be interpreted as an operator in $l^2(G)$ (by left ...
- 6,861
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Amalgamated product of automatic groups
In Gersten's "Problems on Automatic Groups", Problem 14, he asks the following question: Let $G=A\ast_{C}B$ where $A$ and $B$ are automatic and $C$ is infinite cyclic. Is $G$ automatic?
Is this ...
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On the general linear group of a vector space of infinite dimension
Let $F/\mathbb Q$ be a finite normal extension of the rational numbers. Let $V$ be an $F$-vector space of countably infinite dimension, and set $L=GL_F(V)$. Put moreover $L^*$ be the set of all ...
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The finite extensions of $SL_2(q)$ [closed]
Let $G$ be a finite group such that $Z(G)\leq SL_2(q) \leq G $ and $G/Z(G) \cong PGL_2(q)$· Is there any information about the structure of $G$?
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Tarski number is not a quasi isometric invariant, an example?
I know that Tarski number is not a quasi isometric invariant, i.e.
Let $G,H$ be two groups such that $G\sim_{QI} H$, then it is not necessary to have $T(G)=T(H)$.
But can you bring an example for ...
- 885
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How many non-isomorphic groups share the same character table?
I have been thinking for a while about how to enumerate all finite groups. The classical way e.g. here would be to go via latin squares and then try to calculate how many of those obey associativity. ...
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Extension-field subgroups of $\operatorname{GL}(n, K)$
$\newcommand{\op}[1]{\operatorname{#1}}\def\GL{\op{GL}}$I am interested in the subgroups of $\GL(n,K)$ of the form $\GL(n/s, E)$ for $E$ some extension field of $K$ of degree $s$. These arise in the ...
- 9,195
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Relations among hyperplane mirror symmetries
Let $H(n) \subset O(n)$ be the set of mirror symmetries in $\mathbb{R}^n$ with respect to $(n - 1)$-planes containing the origin. One can see that for any $a, b, c \in H(2)$ we have $abcabc = id$, ...
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Do these limits exist?
Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha\in\mathbb C[G]$, the set of complex functions on $G$ with finite support, $\alpha^*\in\mathbb C[G]$ is ...
- 2,118
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Divergence of Groups and Metric Spaces
Several papers, including this and this claim that divergence of finitely generated groups and metric spaces have been introduced by Misha Gromov in his paper "Asymptotic invariants of infinite groups"...
user6976
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On the structure of a finite group of order $144$
Let $G$ be a finite group of order $144$. Suppose there is an irreducible $\mathbb{C}$-character $\theta$ such that $\theta(1)=9$.
QUESTION: Prove $G\cong A_4\times A_4$.
By using Magma, we know ...
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Notation on wreath product [closed]
While reading some (relatively old) papers on group-theory I encountered the following notations whose meanings I cannot understand:
If $W= G \wr H$ is the (unrestricted) wreath product of $G$ and $H$...
6
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1
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402
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Kantor's Singer cycle theorem
I'm trying to understand the proof of Kantor's Singer cycle theorem, which asserts that if $G$ is a subgroup of $\operatorname{GL}(n,q)$ containing a Singer cycle then $\operatorname{GL}(n/s,q^s) \leq ...
- 9,195
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Probability of generation of ${\mathbb Z}^2$
What is the probability that three pairs $(a,b) $ , $(c,d) $ and $(e,f) $ of integers generate $\mathbb Z^2$? As usual the probability is the limit as $n\to \infty$ of the same probability for the $n\...
user6976
4
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What is this group, $A_n = \langle a_1,\ldots a_n \,|\, a_i a_j = a_ja_i \mbox{ if } |i-j| \geq 2\rangle$?
I came across the group with a presentation $A_n = \langle a_1,\ldots a_n \,|\, a_i a_j = a_ja_i \mbox{ if } |i-j| >= 2\rangle$. E.g. $A_1$ and $A_2$ are free groups. Do these groups have a name or ...
- 1,425
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About some positive elements in a group von Neumann algebra
Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha=\sum a_gg$ in $\mathbb C[G]$ (complex functions on $G$ with compact support), $\alpha^*$ is defined to ...
- 2,118
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Branching of anti-symmetric part of product of irreducible representations
This is a corrected version of a previous question (https://mathoverflow.net/questions/303060/branching-of-products?noredirect, suggested for deletion), which contained a wrong conjecture.
Let $V_\...
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Presentation of special linear group over localizations of the integers
I am looking (for $n,k\in{\mathbb Z}$) for a presentation (in the best of all worlds concretely, as a list of relators) for the group ${\rm SL}_n(R)$ for $R={\mathbb Z}[\frac{1}{k}]=\{\frac{a}{k^l}\...
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Does this sequence contain a nonnegative number?
Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha=\sum a_gg$ in $\mathbb C[G]$ (complex functions on $G$ with compact support), $\alpha^*$ is defined to ...
- 2,118
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Group generated by symmetric shears
Consider the multiplicative group generated by matrices of the form
$$
\begin{bmatrix}
{1} & { 0} & { c_1} & {c_3} \\
{0} & {1} & {c_3} & {c_2} \\
{0} & {0} &...
- 1,010
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Dense abstract free subgroups in a free profinite group
Let $\langle a, b \rangle = F_2$ be a two-generator free group and $\hat{F_2}$ be its profinite completion. Is there an element $c\in \hat{F_2}$ such that $\langle a, b, c\rangle \le \hat{F_2}$ is ...
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Complements in $\text{Sub}(\text{Sym}(\omega))$
For any group $G$, we let $\text{Sub}(G)$ be the complete lattice of subgroups of $G$. Let $\text{Sym}(\omega)$ be the group of all bijections $f:\omega\to\omega$.
What is an element of $U\in\text{...
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Is it possible to put Higman group as an amenable by sofic group?
I know Higman group has an amalgamated product decomposition of $BS(1, 2)$. Is it possible to decompose Higman group as some groups we are more familiar with. For example, is there a normal subgroup K ...
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The symmetric group theory of natural numbers
Sometimes it is not easy to formulate a correct question. Here is a better version of this question (I still do not know if it is optimal, but it is better than the previous one).
We say that a set $...
user6976
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Generalizing Autonne-Takagi factorization
Autonne-Takagi factorization (Léon Autonne (1915) and Teiji Takagi (1925)) says that:
A complex symmetric matrix can be 'diagonalized' using a unitary matrix: If $A$ is a rank-$n$ complex symmetric ...
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Quasisimple group with cyclic Sylow p-subgroup and weakly real p-elements?
Does there exist a quasisimple group $G$ and an odd prime $p$ such that $G$ has cyclic Sylow $p$-subgroups and a weakly real element of $p$-power order?
From Strongly real elements of odd order in ...
- 1,070
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Projective resolutions of finite-dimensional representations of infinite groups
Let $G$ be a group and let $V$ be a finite-dimensional complex representation of $G$. Question: Under what circumstances can I find a projective resolution
$$ \cdots \longrightarrow P_3 \...
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About understanding manifold structure on WAP compactification of $\Bbb{C} \rtimes \Bbb{T}$
Let $G$ be a locally compact topological group. A continuous bounded function $f$ on $G$ is called (weakly) almost periodic if the set $L_Gf$ of left translates is relatively compact in the (weak) ...
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How much do we need to add to the generating set of the symplectic group to get $SL(2n,2)$?
Here $Sp(2n,\mathbb{F}_2)$ means the group of matrices preserving the form $\Omega = \left( \begin{array}{cc} 0&I \\ -I&0& \end{array} \right)$, i.e. the symplectic group over an even ...
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$\text{Max}\big(\text{Sub}(\text{Sym}(\omega))\setminus \{\text{Sym}(\omega)\}\big)$
If $G$ is any group, then by $\text{Sub}(G)$ we denote the collection of all subgroups, ordered by $\subseteq$. If $(P,\leq)$ is a partially ordered set we let $\text{Max}(P)$ and the set of maximal ...
- 45.5k
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Fixed point set with non-empty interior
Let $G$ be an infinite compact separable Hausdorff metric group, and $H\subset G$ a closed subgroup, such that the left $G$-action on $G/H$ is effective (i.e., $H$ doesn't contain a non-trivial closed ...
- 1,939
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Characterizing $\mathbf{R}$ as an ordered group
A standard characterization of $\mathbf{R}$ uses the order and the field structure: any linearly ordered field that is archimedean and complete is isomorphic to $(\mathbf{R}, +, \times, <)$ as an ...
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Groups with cyclic radicals
Let $G$ be a torsion-free group. For an element $g \in G \setminus \{1_G\}$ we define the radical of $g$ in $G$ as
$$
Rad_G(g) = \left\{r \in G \mid r^a = g^b \mbox{ for some } a,b \in \mathbb{Z}\...
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Translating first order statements about symmetric groups into the language of numbers and back
A question I was asked recently lead me to the following question. For every closed first order formula $\theta$ in the group signature consider the set $N_\theta$ of natural numbers $n$ such that the ...
user6976
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Counting conjugacy classes with a subgroup of prime index
I am trying to understand the classical method of counting classes from Burnside's old book (Note E) (also clarified a bit by Vera-Lopez, Conjugacy classes in finite solvable groups, 1984) : $G$ is a ...
- 405
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Quotient groups of the lower central series of a surface group
In the answer to MO question 132247, it is possible to find a nice computation of the quotient groups of the lower central series of a finitely generated free group.
Q. What are the quotient ...
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How to show the equality of two descriptions for the cohomology of a non-finite group
I am learning about group cohomology.
For a group $G$ and a $G$-mod $A$, we can define $X^n(G,A)=Map(G^{n+1},A)$, and get a resolution $0\to A\to X^\cdot$ and then define cohomology groups $H^n(G,A)$...
- 806
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The number of distinct closed subgroups of a compact monothetic group
Let $G$ be a connected compact separable Hausdorff metric group, which is monothetic, i.e., has a dense subgroup generated by a single element. Such a group is necessarily Abelian.
Question:
Can the ...
- 1,939
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Counting symmetric subgroups of symmetric groups
This question is related to, but much more specific than, this one.
For $k \leq n$, let $a(k,n)$ denote the number of conjugacy classes of subgroups of the symmetric group $S_n$ which are isomorphic ...
- 2,693
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In 3D point groups, does $[\Gamma_{e}\otimes\Gamma_e] = \Gamma_{Rot_z} \forall$ degenerate $\Gamma_e$ hold in general?
In the following I am referring to groups exclusively describing 3D point symmetries. I use the Schönflies notation for groups and their elements and the Mulliken symbols to describe their irreducible ...