Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
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Triality of Spin(8)
Among simple Lie groups, $Spin(8)$ is the most symmetrical one in the sense that $Out(Spin(8))$ is the largest possible group. A description of this outer automorphism groups is as follows. $Spin(8)$ ...
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Groups in which all characters are rational.
The Symmetric groups $S_n$ has interesting property that all complex irreducible characters are rational (i.e. $\chi(g)\in \mathbb{Q}$ for all $\mathbb{C}$-irreducible characters $\chi$,$\forall g\in ...
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Nonabelian topological fundamental group of a conjugate variety
Let $X$ be a pointed algebraic variety over the field of complex numbers $\mathbb{C}$.
Let $\pi_1^{\rm top}(X)$ and $\pi_1^{\mathrm{\acute{e}t}}(X)$ denote the topological and the étale fundamental ...
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Group with finite outer automorphism group and large center
Does there exist a finitely generated group $G$ with outer automorphism group $\mathrm{Out}(G)$ finite, whose center contains infinitely many elements of order $p$ for some prime $p$?
A motivation is ...
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General Bruhat decomposition (with parabolic not necessarily Borel)
Here is the general Bruhat decomposition (which I have seen in various paper but never with a proof or a complete reference).
Let $G$ be a split reductive group, $T$ a split maximal torus and $B$ a ...
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What are the points of simple algebraic groups over extensions of $\mathbb{F}_1$?
The "field with one element" $\mathbb{F}_1$ is, of course, a very speculative object. Nevertheless, some things about it seem to be generally agreed, even if the theory underpinning them is not; in ...
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element algebraically distinguishable from its inverse
(This question came up in a conversation with my professor last week.)
Let $\langle G,\cdot \rangle$ be a group. Let $x$ be an element of $G$.
Is there always an isomorphism $f : G \to G$ such that ...
user5810
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Can any group be embedded in a simple group?
Any finite group $G$ can be embedded into $A_{|G|+2}$ via Cayley's theorem ($G\hookrightarrow S_{|G|}\hookrightarrow A_{|G|+2}$). If $G$ is not assumed to be finite, is it still always possible to ...
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Can a group be a finite union of (left) cosets of infinite-index subgroups?
To be more precise (but less snappy): is there an example of a group $G$ with finitely many infinite-index subgroups $H_1,\dots, H_n$ and elements $k_1,\dots, k_n$ such that $G$ is the union of the ...
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Nilpotency of a group by looking at orders of elements
For any finite group $G$, let
$$\theta(G) := \sum_{g \in G} \frac{o(g)}{\phi(o(g))},$$
where $o(g)$ denotes the order of the element $g$ in $G$, and where $\phi$ is the Euler totient function.
It is ...
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Periodic Automorphism Towers
In Scott's classic textbook on Group Theory, he asks:
Suppose that $G$ is a finite group. Is the sequence of isomorphism types
of the groups $Aut^{(n)}(G)$ for $n \in \mathbb{N}$ eventually periodic?
...
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Finite groups with the same character table
Say I have two finite groups $G$ and $H$ which aren't isomorphic but have the same character table (for example, the quaternion group and the symmetries of the square). Does this mean that the ...
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Combinatorial Techniques for Counting Conjugacy Classes
The number of conjugacy classes in $S_n$ is given by the number of partitions of $n$. Do other families of finite groups have a highly combinatorial structure to their number of conjugacy classes? For ...
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Criteria for irreducibility of polynomial
If $f, g\in \mathbb C[a,b]$ are polynomials in two variables, are there easy criteria that allow to see if $f(x,y)-g(t,z)\in \mathbb C[x,y,t,z]$ is irreducible?
Thank you very much,
best
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When is the torsion subgroup of an abelian group a direct summand?
For an abelian group $G$, let $G[\operatorname{tors}]$ be its torsion subgroup.
Consider the torsion sequence:
$0 \rightarrow G[\operatorname{tors}] \rightarrow G \rightarrow G/G[\operatorname{tors}] \...
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Are almost commuting hermitian matrices close to commuting matrices (in the 2-norm)?
I consider on $M_n(\mathbb C)$ the normalized $2$-norm, i.e. the norm given by $\|A\|_2 = \sqrt{\mathrm{Tr}(A^* A)/n}$.
My question is whether a $k$-uple of hermitian matrices that are almost ...
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Is there a "universal group object"? (answered: yes!)
I want to say that a group object in a category (e.g. a discrete group, topological group, algebraic group...) is the image under a product-preserving functor of the "group object diagram", $D$. One ...
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Proofs of the Stallings-Swan theorem
It is a well-known and deep${}^\ast$ theorem that if a group $G$ has cohomological dimension one then it must be free. This was proved in the late 60's by Stallings (for finitely generated groups) and ...
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Word maps on compact Lie groups
Let $w=w(a,b)$ be a non-trivial word in the free group $F_2 = \langle a,b \rangle$ and $w_G \colon G \times G \to G$ be the induced word map for some compact Lie group $G$.
Murray Gerstenhaber and ...
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Non-residually finite matrix groups
By Malcev's theorem, every finitely generated linear group is residually finite (RF).
On the other hand, say, the group of rational numbers is linear, but is not residually finite. Thus, one has to ...
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Generation of finite index subgroups
Related to a question by Mark Sapir (see here) and a question by Kate Juschenko (see here), let me ask the following:
Question: Let $G$ be a finitely generated group and let $\varepsilon>0$. Is ...
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Generation in a group versus generation in its abelianization.
Background
I have been spending a lot of time in my research with subsets of groups that are very close to being generating sets. To make this precise:
Let $G$ be a group. If a subset $S$ of $G$ ...
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How many elements does it take to normally generate a group?
$\DeclareMathOperator\nr{nr}\DeclareMathOperator\rank{rank}$This is a terminology question (I should probably know this, but I don't). Given a group $G$, consider the minimal cardinality $\nr(G)$ of a ...
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Does the hypergraph of subgroups determine a group?
A hypergraph is a pair $H=(V,E)$ where $V\neq \emptyset$ is a set and $E\subseteq{\cal P}(V)$ is a collection of subsets of $V$. We say two hypergraphs $H_i=(V_i, E_i)$ for $i=1,2$ are isomorphic if ...
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Row of the character table of symmetric group with most negative entries
The row of the character table of $S_n$ corresponding to the trivial representation has all entries positive, and by orthogonality clearly it is the only one like this.
Is it true that for $n\gg 0$, ...
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Estimate for the order of the outer automorphism group of a finite simple group
It is known (given CFSG) that all non-abelian finite simple groups have small outer automorphism groups. However, it's quite tedious to list all the possibilities. Does anyone know a reference for a ...
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Finitely presented sub-groups of $\operatorname{GL}(n,C)$
Here are two questions about finitely generated and finitely presented groups (FP):
Is there an example of an FP group that does not admit a homomorphism to $\operatorname{GL}(n,C)$ with trivial ...
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The finite groups with a zero entry in each column of its character table (except the first one)
$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Aut{Aut}$Consider the class of finite groups $G$ having a zero entry in each column of its character table (except the first one), i.e. for all $g \...
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Relationship between the cohomology of a group and the cohomology of its associated Lie algebra
Let $G$ be a group and let $k$ be a field (characteristic 0 if you want). Let $L$ be the graded Lie ring associated to the lower central series of $G$, that is, $L$, as a graded abelian group is $\...
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How small can a group with an n-dimensional irreducible complex representation be?
More precisely, what is the smallest exponent e such that, for every n, there exists a group of size at most Cn^e for some absolute constant C and with an n-dimensional irreducible complex ...
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(weak?) BN-Pair / Tits System for Sporadic Groups
The structure of finite simple groups of Lie type of arbitrary rank can be described well via BN-pairs. BN-pairs basically generalize the Bruhat decomposition of matrices into monomial $N$ and ...
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A character identity
This is related to my question, but it concerns a specific point of the proof of Schur's Theorem.
Let $G$ be a finite group and $\chi$ an irreducible character of $G$. Is it true that
$$\forall g\in ...
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How is this group theoretic construct called?
Let $G$ be a finite group, $S\subset G$ a generating set, $|g| = |g|_S = $ word length with respect to $S$. Define the "defect" of $g,h$ to be
$$\psi(g,h) = |g|+|h|-|gh|$$
Then $\psi:G\times G \...
user6671
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Without choice, can every homomorphism from a profinite group to a finite group be continuous?
In ZFC, some homomorphisms from profinite groups to finite groups are discontinuous. For instance, see the examples in this question. However, all three constructions given use consequences of the ...
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Are there Hamilton paths in Cayley graphs of Coxeter groups?
Hi everyone.
I want to optimize certain computation on finite Coxeter groups $(W,S)$. Basically I compute the matrices $\rho(T_w)$ for all $w\in W$ of a matrix representation $H\to K^{d\times d}$ of ...
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Are extensions of linear groups linear?
A group $G$ is said to be linear if there exists a field $k$, an integer $n$ and an injective homomorphism $\varphi: G \to \text{GL}_n(k).$
Given a short exact sequence
$1 \to K \to G \to Q \to 1$ ...
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For which rings $R$ is $\mathrm{SL}_n(R)$ generated by transvections?
Let $R$ be a commutative ring $R$ with $1$, and $n \geq 2$ an integer.
Under which conditions is the group $\operatorname{SL}_n(R)$ generated by transvections?
(A transvection is a matrix with $1$ ...
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Is Hopf property a quasi-isometry invariant?
Recall that a group $G$ is called Hopfian if every surjective endomorphism $G\to G$ is injective. Malcev observed that all finitely-generated (f.g.) residually finite groups are Hopfian. It is well-...
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Is a retract of a free object free?
I wonder whether this is true in the categories of groups, monoids, commutative algebras, associative algebras, Lie algebras?
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Proper discontinuity and existence of a fundamental domain
I am currently teaching a topics course where I talk about some discrete groups acting properly. A student asked a very basic question that stumped me: what is the precise relationship between proper ...
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What are the auto-equivalences of the category of groups?
My question is motivated by Are the inner automorphisms the only ones that extend to every overgroup?
What are the auto-equivalences of the category of groups? What kind of structure do they form?
...
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In a compact lie group, can two closed connected subgroups generate a non-closed subgroup?
Let $H$,$K$ be closed connected subgroups of a compact Lie group $G$. Let $L:=\langle H,K \rangle$ be the subgroup they generate, ie, the smallest subgroup of $G$ containing them both. Must $L$ be ...
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Why do these two Monster-related calculations yield $163$?
Fact 1: (1979, Conway and Norton)$^{1}$
"There are $194-22-9=\color{blue}{163\,}$ $\mathbb{Z}$-independent McKay-Thompson series for the Monster."
Note: There are 194 (linear) irreducible ...
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Number of solutions to equations in finite groups
Suppose $G$ is a finite group and that $E$ is an equation of the form $x_1 x_2 ... x_n = e$, where each $x_i$ is in the set of symbols $\{x, y, x^{-1}, y^{-1}\}$.
Is it always true that the number ...
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Which commutative groups are the group of units of some field?
Inspired by a recent question on the multiplicative group of fields. Necessary conditions include that there are at most $n$ solutions to $x^n = 1$ in such a group and that any finite subgroup is ...
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Algorithm to test for discrete or quasi-Fuchsian subgroups of PSL(2,C)
Let $\Gamma = \pi_1(S)$ denote the fundamental group of a compact surface $S$ of genus $g>1$.
Given a representation $\rho : \Gamma \to \mathrm{PSL}(2,\mathbb{C})$, specified by matrix ...
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Finitely generated group with $\aleph_0<X_G<2^{\aleph_0}$ normal subgroups?
Let $X_G$ be the number of normal subgroups of a group $G$. Are there examples of finitely generated groups $G$ where it is consistent to have $\aleph_0<X_G<2^{\aleph_0}$ normal subgroups? Also ...
user35370
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Cohomological dimension of $G \times G$
$\DeclareMathOperator\cd{cd}$A question that I have already posted in the Mathematics section, but which seems to be too delicate for that section (see here and here):
Let $\cd(G)$ denote the ...
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Mapping class group and property (T) [closed]
Does anyone know what the current expert consensus is concerning the status of the question as to whether the mapping class group of a surface has property (T)?
There is a short (21 page) paper by J. ...
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fundamental groups of complements to countable subsets of the plane
This question is a follow-up of this MSE post and a comment by Henno Brandsma:
Question 1. Let $S$ be the set of isomorphism classes of fundamental groups $\pi_1(E^2 - C)$, where $C$ ranges over all ...
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