Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
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Are the homogeneous single chain subfactors, Dedekind?
Background: See here and there.
Recall that a subfactor is Dedekind if all its intermediate subfactors are normal.
A subfactor $(N \subset M)$ is Homogeneous Single Chain (HSC) if its lattice ...
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Can every cancellative invertible-free monoid be embedded in a group?
A monoid is invertible-free if $xy=1$ implies $x=y=1$ for all $x,y$.
Question: Can every cancellative invertible-free monoid be embedded in a group?
I'm fairly sure that a quotient of the free product ...
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Normal intermediate subgroup and normal core
Let $G$ be a finite group and $H$ a subgroup.
The normal core of $H$ in $G$ is $core_G(H) := \bigcap_{g \in G}g^{-1}Hg$
Definition: $K$ is a normal intermediate subgroup of the inclusion $(H \subset ...
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Existence of a multiplication bifunctor for the category of groups
For $\mathsf{Grp}$ the category of groups, a bifunctor $M: \mathsf{Grp} \times \mathsf{Grp}\to \mathsf{Grp}$ is a multiplication bifunctor if:
$M(C_n,C_m) \simeq C_{nm}$,
$M(C_1,G) \simeq M(G,C_1) \...
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Existence of an addition bifunctor for the category of groups
Let $\mathsf{Grp}$ be the category of groups. A bifunctor $A: \mathsf{Grp} \times \mathsf{Grp}\to \mathsf{Grp}$ is an addition bifunctor if:
$A(C_n,C_m) \simeq C_{n+m}$,
$A(C_0,G) \simeq A(G,C_0) \...
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Abelian subfactors, a relevant concept?
Through the questions below, this post asks whether the concept of abelian subfactor is relevant.
Remark : here abelian qualifies an inclusion of II$_1$ factors $(N \subset M)$, $N$ is not an abelian ...
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Minimal number of defining relators of a finite $p$-group on a minimal generating set
What is the state-of-art of the following question?
Let $p$ be any prime number. For any finite $p$-group $G$, let $r_G$ denote the minimum number of defining relators in all presentations of $G$ ...
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$SO(3)$ 2-cocycle trivialized to a 2-coboundary in $SU(2)$?
I was trying to understand this interesting question by example.
Let me follow their previous discussion and ask: Let a generic nontrivial 2-cocycle $\omega_2^G(g_1,g_2) \in H^2(G,\mathbb{R}/\mathbb{...
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About the conjugation of semi-simple subgroups
Let $G$ be a semi-simple algebraic group over $\mathbb{Q}$, I would like to find an integer $d>0$ only depending on $G$ with the following property. For any two semi-simple $\mathbb{Q}$-subgroups $...
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A second isomorphism theorem for the inclusions of groups
The usual second isomorphism theorem for groups is: let $G$ be a group, $S$ and $N$ subgroups with $N$ normal, then $SN$ is a subgroup of $G$, $S\cap N$ is a normal subgroup of $S$ and $SN/N \simeq ...
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Roots of permutations
Consider the equation $x^2=x_0$ in the symmetric group $S_n$, where $x_0\in S_n$ is fixed. Is it true that for each integer $n\geq 0$, the maximal number of solutions (the number of square roots of $...
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Conceptual reason why the sign of a permutation is well-defined?
Teaching group theory this semester, I found myself laboring through a proof that the sign of a permutation is a well-defined homomorphism $\operatorname{sgn} : \Sigma_n \to \Sigma_2$. An insightful ...
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Group cohomology and condensed matter
I am mystified by formulas that I find in the condensed matter literature
(see Symmetry protected topological orders and the group cohomology of their symmetry group arXiv:1106.4772v6 (pdf) by Chen, ...
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Why can't a nonabelian group be 75% abelian?
This question asks for intuition, not a proof.
An earlier question,
Measures of non-abelian-ness
was thoroughly answered by Arturo Magidin.
A paper by Gustafson1
proves that, for a nonabelian group,
...
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Is Thompson's Group F amenable?
Last year a paper on the arXiv (Akhmedov) claimed that Thompson's group $F$ is not amenable, while another paper, published in the journal "Infinite dimensional analysis, quantum probability, and ...
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Is there an odd-order group whose order is the sum of the orders of the proper normal subgroups?
For a finite group G, let |G| denote the order of G and write $D(G) = \sum_{N \triangleleft G} |N|$, the sum of the orders of the normal subgroups. I would like to call G "perfect" if D(G) = 2|G|, ...
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Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?
$\DeclareMathOperator\SO{SO}\newcommand{\R}{\mathbb{R}}\newcommand{\S}{\mathbb{S}}$The periodic table of elements has row lengths $2, 8, 8, 18, 18, 32, \ldots $, i.e., perfect squares doubled. The ...
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Can we ascertain that there exist an epimorphism $G\rightarrow H?$
Let $G,H$ be finite groups. Suppose we have a epimorphism $G\times G\rightarrow H\times H$. Can we find an epimorphism $G\rightarrow H$?
A fellow graduate student asked me this question during TA ...
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Transitivity on $\mathbb{N}_0$ -- a 42 problem
Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$.
Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class
transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the ...
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Collapsible group words
What is the length $f(n)$ of the shortest nontrivial group word $w_n$ in $x_1,\ldots,x_n$ that collapses to $1$ when we substitute $x_i=1$ for any $i$?
For example, $f(2)=4$, with the commutator $[...
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Square roots of elements in a finite group and representation theory
Let $G$ be a finite group. In an an earlier question, Fedor asked whether the square root counting function $r_2:G\rightarrow \mathbb{N}$, which assigns to $g\in G$ the number of elements of $G$ that ...
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Feit-Thompson theorem: the Odd order paper
For reference, the Feit-Thompson Theorem states that every finite group of odd order is necessarily solvable. Equivalently, the theorem states that there exist no non-abelian finite simple groups of ...
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The finite subgroups of SL(2,C)
Books can be written about the finite subgroups of $\mathrm{SL}(2,\mathbb C)$ (and their immediate family, like the polyhedral groups...) I am about to start writing notes for a short course about ...
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Are the inner automorphisms the only ones that extend to every overgroup?
Let $H$ be a group. Can we find an automorphism $\phi :H\rightarrow H$ which is not an inner automorphism, so that given any inclusion of groups $i:H\rightarrow G$ there is an automorphism $\Phi: G\...
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A “mother of all groups”? What kind of structures have "mother of all"s?
For ordered fields, we have a “mother of all ordered fields”, the surreal numbers $\mathbf{No}$, a proper-class “field” which includes (an isomorphic copy of) every other ordered field as a subfield. ...
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Is it decidable to check if an element has finite order or not?
Suppose we have a finitely presented group $G$ with decidable word problem. Is it decidable to check whether a given element $x\in G$ has finite order or infinite?
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When is Aut(G) abelian?
Let $G$ be a group such that $\operatorname{Aut}(G)$ is abelian. Is then $G$ abelian?
This is a sort of generalization of the well-known exercise, that $G$ is abelian when $\operatorname{Aut}(G)$ is ...
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Does the hypergraph structure of the set of subgroups of a finite group characterize isomorphism type?
Question
Suppose there is a bijection between the underlying sets of two finite groups $G, H$, such that every subgroup of $G$ corresponds to a subgroup of $H$, and that every subgroup of $H$ ...
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What is the difference between PSL_2 and PGL_2?
Let $K$ be a field and $G:=SL_2(K)$, then $G$ is a $K-$split reductive group (to use some big words). These groups are classified by a based root datum $(X,D,X',D')$. Let $G'$ be group associated to $(...
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Action of PGL(2) on Projective Space
Let $k$ be a field, let $G = PGL_2(k)$ be the projective general linear group of $k$, and let
$X = k \cup \{ \infty \}$ be one-dimensional projective space over $k$. Then $G$ acts on $X$ (via ...
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Applications of infinite graph theory
Finite graph theory abounds with applications inside mathematics itself, in computer science, and engineering. Therefore, I find it naturally to do research in graph theory and I also clearly see the ...
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Conjugacy classes of $\mathrm{SL}_2(\mathbb{Z})$
I was wondering if there is some description known for the conjugacy classes of $$\mathrm{SL}_2(\mathbb{Z})=\{A\in \mathrm{GL}_2(\mathbb{Z})|\;\;|\det(A)|=1\}.$$ I was not able to find anything about ...
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A profinite group which is not its own profinite completion?
Is there a profinite group $G$ which is not its own profinite completion?
Surely not, I thought. But upon looking into it, I found that there is a special name given to a $G$ which is its own ...
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Do the Baumslag-Solitar groups occur in nature?
The
Baumslag-Solitar groups
$BS(m,n) = \langle b,s\mid s^{-1}b^ms = b^n\rangle$, with $mn\neq 0$,
are important examples (more often, counter-examples) in group theory.
They are residually finite if, ...
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What is the defining property of reductive groups and why are they important?
Having read (skimmed more like) many surveys of the Langlands Program and similar, it seems the related ideas apply exclusively to groups that are "reductive".
But nowhere, either in these surveys or ...
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The field of fractions of the rational group algebra of a torsion free abelian group
Let $G$ be a torsion free abelian group (infinitely generated to get anything interesting). The group algebra $\mathbb{Q}[G]$ is an integral domain. Let $\mathbb{Q}(G)$ be its field of fractions.
...
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Counting the Groups of Order n Weighted by 1/|Aut(G)|
Idle question:
Let $g(n)$ be the sum, over all isomorphism classes of groups of order $n$, of $\frac{1}{|Aut(G)|}$ where $G$ is a group in the class. Thus $g(n)n!$ is the number of group laws on a ...
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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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Classifying Space of a Group Extension
Consider a short exact sequence of Abelian groups -- I'm happy to assume they're finite as a toy example:
$$
0 \to H \to G \to G/H \to 0\ .
$$
I want to understand the classifying space of $G$. Since ...
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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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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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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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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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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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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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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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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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