Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,181 questions
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Subgroups of free abelian groups are free: a topological proof?
There is a well-known topological proof of the fact that subgroups of free groups are free. Many people, myself included, think it is easier and more natural than the purely algebraic proofs which ...
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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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Why can the general quintic be transformed to $v^5-5\beta v^3+10\beta^2v-\beta^2 = 0$?
The quintic can be transformed to the one-parameter Brioschi quintic,
$$u^5-10\alpha u^3+45\alpha^2u-\alpha^2 = 0\tag1$$
This form is well-known for its connection to the symmetries of the ...
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Number of subgroups in a Bieberbach group.
Assume $\Gamma$ be a Bieberbach group which acts on $\mathbb R^n$
(i.e. a discrete subgroup of isometries of $n$-dimensional Euclidean
space with a compact fundamental domain).
Denote by $M(\Gamma)$ ...
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Does any textbook take this approach to the isomorphism theorems?
Below, I present an outline of a proof of the first isomorphism theorem for groups. This is how I usually think of the first isomorphism theorem for ______________, but groups will get the points ...
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Uniform proof that a finite (irreducible real) reflection group is determined by its degrees?
Given a finite (irreducible real) group $G$ generated by reflections acting on euclidean $n$-space, it was shown by Chevalley in the 1950s that the algebra of invariants of $G$ in the associated ...
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The congruence subgroup property for mapping class groups and a conjecture of Grothendieck
This question is about a link between an open question in low-dimensional topology and a conjecture of Grothendieck, proved by Mochizuki. Let's start by stating them.
Recall that a subgroup $K$ of a ...
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Universal group?
I can construct a finitely presented group $G$ with the following property (which I use to construct something else).
Given a finitely preseted group $\Gamma$, there is a subgroup $G'\le G$ of finite ...
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Groups whose finite index subgroups of fixed index are isomorphic
I am interested in finitely generated groups $G$ that are residually finite and have the following property: For each $d \geq 1$, $G$ has subgroups of finite index $d$, and all such subgroups are ...
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Are amenable groups topologizable?
I've learned about the notion of topologizability from "On topologizable and non-topologizable groups" by Klyachko, Olshanskii and Osin (http://arxiv.org/abs/1210.7895) - a discrete group $G$ is ...
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Statements in group theory which imply deep results in number theory
Can we name some examples of theorems in group theory which imply (in a relatively straight-forward way) interesting theorems or phenomena in number theory?
Here are two examples I thought of:
The ...
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Are infinite groups in which most elements have order $\leq 2$ commutative?
The starting point of this question is the following:
If $G$ is a group such that all elements have order at most $2$, then $G$ is commutative.
If $G$ is any group, let $G_{>2}$ denote the set ...
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Non-vanishing of group cohomology in sufficiently high degree
Atiyah in his famous paper , Characters and cohomology of finite groups, after proving completion of representation ring in augmentation ideal is the same as $ K(BG)$, gives bunch of corollaries of ...
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Is a quotient of a reductive group reductive?
Is a quotient of a reductive group reductive?
Edit [Pete L. Clark]: As Minhyong Kim points out below, a more precise statement of the question is:
Is the quotient of a reductive linear group by a ...
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Faithful representations and tensor powers
The following result was mentionned earlier in this thread, I searched a bit in the related threads and couldn't find a proof. I would really like to see a proof of it:
Let $G$ be a finite group and $...
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Why are subfactors interesting?
I get asked this question a lot, and am not very happy with any of the answers.
Vaguely I think of subfactor theory as a generalization of representation theory of groups. That is, if you have a ...
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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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Subgroups of GL(2,q)
I have been wondering about something for a while now, and the simplest incarnation of it is the following question:
Find a finite group that is not a subgroup of any $GL_2(q)$.
Here, $GL_2(q)$ is ...
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Enumeration of a finite group
Let $G=\{g_1,g_2,...,g_n\}$ be a group with $e=g_1$ and $n$ is odd,
Set $$a_1=g_1$$
$$a_2=g_1g_2$$
$$a_3=g_1g_2g_3$$
$$a_n=g_1g_2...g_n$$
I am looking for example that all $a_i$ are different from ...
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Why is $(\mathbb{Z}/3\mathbb{Z})^3$ not a class group of an imaginary quadratic number field ?
In general, it seems not known which finite abelian groups are class groups of quadratic number fields.
For imaginary quadratic number fileds, I read that $(\mathbb{Z}/3\mathbb{Z})^3$ is the smallest ...
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Is a group uniquely determined by the sets $\{ab,ba\}$ for each pair of elements a and b?
This is a cross-posted question, originally active here on math.stackexchange.
For a given group $G=(S,\cdot)$ with underlying set $S$, consider the function
$$
F_G:S\times S\to\mathcal P(...
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Solvable groups that are linear over $\mathbb{C}$ but not over $\mathbb{Q}$?
Let $\Gamma$ be a finitely generated finitely presented virtually solvable group. Assume that there exists an injective representation $\Gamma \to \operatorname{GL}_n(\mathbb{C})$. Is it true that ...
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divisors of $p^4+1$ of the form $kp+1$
In group theory the number of Sylow $p$-subgroups of a finite group $G$, is of the form $kp+1$.
So it is interesting to discuss about the divisors of this form. As I checked it seems that for an odd ...
- 1,168
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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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Finitely presented infinite group with no element of infinite order?
Is there an example of a finitely presented infinite group in which every element has finite order? Or, is it known that every finitely presented infinite group has an element of infinite order?
I ...
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Orbit structures of conjugacy class set and irreducible representation set under automorphism group
let G be a finite group. Suppose C is the set of conjugacy classes of G and R is the set of (equivalence classes of) irreducible representations of G over the complex numbers.
The automorphism group ...
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Categorical Unification of Jordan Holder Theorems
In addition to the Jordan-Holder theorem for groups, there are various Jordan-Holder Theorems for other categories:
Finite dimensional representations have filtrations whose associated graded ...
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What methods exist to prove that a finitely presented group is finite?
Suppose I have a finitely presented group (or a family of finitely presented groups with some integer parameters), and I'd like to know if the group is finite. What methods exist to find this out? I ...
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Realizing groups as commutator subgroups
What are the groups $X$ for which there exists a group $G$ such that $G' \cong X$?
My considerations:
$\bullet$ If $X$ is perfect we are happy with $G=X$.
$\bullet$ If $X$ is abelian then $G := X \...
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Modern references on hyperbolic groups
Several good references dedicated to hyperbolic groups have been written until 1990, including:
Hyperbolic groups, written by M. Gromov.
Géométrie et théorie des groupes : les groupes hyperboliques ...
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products of conjugates in free groups
While trying to carry out some technical arguments in free groups, I have encountered the following problem, to which I don't know the answer.
Let $F$ be a free group and let $g,a_1,\ldots,a_n \in F$....
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What is a Gaussian measure?
Let $X$ be a topological affine space. A Gaussian measure on $X$ is characterized by the property that its finite-dimensional projections are multivariate Gaussian distributions.
Is there a direct ...
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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}] \...
- 65.4k
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Properties of a non-sofic group
This question is, essentially, a comment of Mark Sapir. I think it deserves to be a question.
A countable, discrete group $\Gamma$ is sofic if for every $\epsilon>0$ and finite subset $F$ of $\...
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Does the 3875-dimensional rep of $E_8$ have a solution to $x\star x=0$?
Consider the compact Lie group $E_8$. Its second-smallest fundamental representation is $3875$-dimensional and admits a symmetric invariant form, and so is real: $E_8 \curvearrowright \mathbb{R}^{3875}...
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Linear groups that are nonlinear over the integers
What are sources of finitely generated $\mathbb C$-linear groups that are not $\mathbb Z$-linear?
Recall that a group is $R$-linear if it is isomorphic to a subgroup of $GL(n,R)$ for some $n$, where $...
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In an inductive family of groups, does the probability that a particular word is satisfied converge?
We have some group word $w$ in $k$ letters. We say a $k$-tuple of group elements $\vec{g} = (g_1, g_2, \ldots , g_k) \in G^k$ satisfies the word $w$ if $w$ gives the identity at $\vec{g}$. More ...
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Kaplansky's unit conjecture and unique products
There are three conjectures on group rings that bear the name of Kaplansky (see for example this question). The 'unit conjecture' in the title of the present question is the strongest of them, and ...
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Do all possible trees arise as orbit trees of some permutation groups?
I.Motivation from descriptive set theory
(Contains some quotes from Maciej Malicki's paper.)
The classical theorem of Birkhoff-Kakutani implies that every metrizable topological group G admits a ...
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Is there a "crash-course" book on Abelian varieties (e.g., an introduction for physicists)?
Hello,
In our (rather applied) theoretical physics research, we have encountered an important class of problems, which seem to require an understanding of Abelian functions (unfortunately, this ...
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Homomorphism from $\hat{\mathbb{Z}}$ to $\mathbb{Z}$
I expect this question has a very simple answer.
We all know from primary school that there are no non-trivial continuous homomorphisms from $\hat{\mathbb{Z}}$ to $\mathbb{Z}$. What if we forget ...
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When is a extension of $\mathbb{Z}$ by a free group a CAT(0) group?
The question has an easy answer, if one replaces free by free abelian: Then the resulting group is always solvable and a solvable subgroup of a CAT(0) group is virtually abelian.
If the resulting was ...
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Are all free groups linear, i.e., admit a faithful representation to GL(n,K) for some field K ?
All free groups of finite or infinite countable rank are subgroups of the free non-abelian group $F_2$, which is linear. However, a free group of infinite uncountable rank will not be a subgroup of $...
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maximal order of elements in GL(n,p)
I am looking for a formula for the maximal order of an element in the group $\operatorname{GL}\left(n,p\right)$, where $ p$ is prime.
I recall seeing such a formula in a paper from the mid- or early ...
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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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Is any interesting question about a group G decidable from a presentation of G?
We say that a group G is in the class Fq if there is a CW-complex which is a BG (that is, which has fundamental group G and contractible universal cover) and which has finite q-skeleton. Thus F0 ...
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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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The role of the Automatic Groups in the history of Geometric Group Theory
What is the role of the theory of Automatic Groups in the history of Geometric Group Theory?
Motivation:
When I read through the "Word Processing in Groups" I was amazed by the supreme beauty and ...
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On permanents and determinants of finite groups
$\DeclareMathOperator\perm{perm}$Let $G$ be a finite group. Define the determinant $\det(G)$ of $G$ as the determinant of the character table of $G$ over $\mathbb{C}$ and define the permanent $\perm(G)...
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Mapping from a finite index subgroup onto the whole group
Dear All,
here is the question:
Does there exist a finitely generated group $G$ with a proper subgroup $H$ of finite index, and an (onto) homomorphism $\phi:G\to G$ such that $\phi(H)=G$?
My guess ...
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