Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,182 questions
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Cyclic subgroups of finite $p$-groups
Let $G$ be a finite non-Dedekind $p$-group with non-cyclic center, where $p$ is an odd prime.
By $[\langle x\rangle]_G=\{g^{-1}\langle x\rangle g\ |\ g\in G\},$
I mean the conjugacy class of the ...
- 487
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Deduce Schur-Zassenhaus theorem from Wedderburn-Malcev theorem
Is it possible to derive Schur-Zassenhaus theorem from Wedderburn-Malcev theorem? For the existance part I have the following idea: Let $G$ be a finite Group and $N$ an abelian Hall $p$-subgroup. Let $...
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Counting elements having a given cycle structure in maximal subgroups of a generalized symmetric group
Let $G$ be the wreath product $C_7\wr S_{18}$, where $C_7$ is the cyclic group of order 7 and $S_{18}$ is the symmetric group on 18 symbols. Consider $G$ to be embedded in the group $S_{126}=S_{7\cdot ...
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Automorphism groups of graphs of bounded treewidth
The celebrated Frucht's theorem states that every finite group is isomorphic to the automorphism group of a finite graph $G$. If we restrict $G$ to belong to a certain class, some groups may become ...
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epimorphisms between perfect groups
Is it true that for every infinite perfect group $P$, there exists a group $G$ with the following conditions:
$G$ and $P$ have the same cardinality;
$P$ is isomorphic to a quotient of $G$;
$G$ is not ...
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graphs with semiregular automorphisms
I need some "well-known" non-regular finite graphs (at least two vertices have different valency) whose automorphism groups contain a non-trivial subgroup that acts on the vertices semi-regularly (i....
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Characterisation of supersolvability of a finite group
Definition 1: A subgroup $H$ of a group $G$ is said to be abnormal in $G$ if for each $g\in G$, we have $g\in \langle H, H^g \rangle$.
Definition 2: A finite group $G$ is called a $B$-group if every $...
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Does a quotient group $G/N$ have a natural action on the regular representation of $G$?
Let $G$ be a group. I am happy to assume niceties such as finite and abelian, but perhaps it is not necessary to answer my question.
Consider the $|G|$-dimensional vector space $V$ (over some nice ...
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Characterisation of finite solvable T-group
Definition: A $T$-group is a group in which normality is a transitive relation.
Definition: A subgroup $H \leq G$ is said to be weakly normal in $G$ if for each $g\in G$, $H^g \leq N_G(H)$ implies ...
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Minimal normal subgroup of a finite perfect group
Suppose $G$ is a finite perfect group, $N$ is an Abelian minimal normal subgroup of $G$ and $$G/N=SL_2(q),$$ where $q=2^f$ for some integer $f\geq5$.
What can we say about the order of $N$?
Thanks!
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non-solvable finite group with a certain character degrees
Is there any non-solvable finite group $G$ such that ${\rm cd}(G) =\{1, 3, 4, 5, 6, 10 \}$, where ${\rm cd}(G)$ is the set of irreducible character degrees of $G$. In other words we have that a ...
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Intersection of thickly syndetic sets
Question: Let $\Gamma$ be a countable group. Is the intersection of two thickly syndetic sets still thickly syndetic?
I've only seen the proof for the group $\mathbb{Z}$ (and I believe this method ...
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Factoring in discrete Heisenberg group $H_3(\mathbb{Z})$
Let $H_3(\mathbb{Z})$ be the discrete Heisenberg group generated by $x=\begin{pmatrix}
1 & 1 & 0\\
0 & 1 & 0\\
0 & 0 & 1\\
\end{pmatrix},\ \ y=\begin{pmatrix}
1 & 0 &...
user6671
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Classification of transitive subgroups of finite symmetric groups generated by double transpositions
I want to classify (up to isomorphism) all transitive subgroups of symmetric group $S_n$ which are generated by double transpositions (product of two transpositions). Is there a characterization for ...
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central automorphisms
Let $C_{Aut_{c}(G)}(Z(G))$ be the group of all central automorphisms of finite non abelian $p$-group $G$ fixing $Z(G)$ element wise. If $C_{Aut_{c}(G)}(Z(G))$ is a proper subgroup ...
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A question about Mostow's theorem for self-adjoint groups
In Mostow's paper Self-adjoint groups, one can find the following property.
Theorem. Let $G\subset \mathrm{GL}_{n,\mathbb{R}}$ be a reductive real algebraic subgroup. Then there exists $a\in \...
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Symmetric analogue of "alternating bihomomorphism is skew of 2-cocycle" theorem
Let $G$ be a finite abelian group. It is well-known that every alternating bihomomorphism $\Omega:G\times G \to \mathbb{C}^\times$ arises as the skew $\kappa/\kappa^T$ of a 2-cocycle $\kappa \in Z^2(G,...
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Inverse limits and first isomorphism theorem for compact topological groups
This question was originally asked on MathSE here.
I have a problem with Proposition (1.2.1) from J. Wilson's book 'Profinite Groups'
The proposition is the following:
Let $(G, \varphi_i : G \to ...
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Non-existence of nontrivial finite group extension of any simply-connected Lie group
Let $Q$ be a simply-connected compact Lie group. Can one outline the proof (or provide the counter examples if my statement is false) that
there does not exist any group $G$ (with no topology) ...
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Asymptotic cone of discrete group of Heisenberg group $\mathbb{H}^3$
Note that $(\mathbb{Z}^2,d_W)$ where $d_W$ is word metric has asymptotic cone $$(\mathbb{R}^2,\| \ \|_1)=\lim_{t>0\rightarrow 0}\ t(\mathbb{Z}^2,d_W)$$
And Heisenberg group $\mathbb{H}^3$ has an ...
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Cocompact (finite covolume) lattices in euclidean groups
1) Is there a classification of cocompact ( or finite co-volume) lattices in Euclidean groups E(n)( motions of Euclidean space) ( especially in dimensions 2,3,4)?
2) Also what is (if any) the ...
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Fuchsian groups and surface groups
The following question may be trivial or inappropriate; I am not sure though.
It is known that a cocompact oriented Fuchsian group $\Gamma$ admits a presentation: for given $m,g,d_i \geq 0$
$$
\...
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A question about Henry Wiltons paper "Hall's Theorem for limit groups"
I have some question about the last conclusion in the proof of Lemma 2.16 on page 20 of the paper "Hall's theorem for limit groups". You can find the paper on
https://arxiv.org/abs/math/0605546
So. ...
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We know $A_5$ as a non-CI-group. Now, is $A_5$ a BI-group?
We call a group satisfying the following property for all $\nu \in cd(G)$ (Irreducible character degrees of $G$) a BI-group (Babai Invariant group)
Let $G$ be a finite group, let $\Gamma=Cay(G,S)$...
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Reference request for the list of nilpotent subgroups of SU(2)?
It's not hard to show that all non-abelian nilpotent subgroups of $SU(2)$ are actually finite and in fact are conjugate to one of the generalized quaternion groups of order a power of two, $$Q_{2^n} =...
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How quickly can one compute the Hurwitz action of braid groups on finite groups?
Let $G$ be a finite group. Define the Hurwitz action of $B_{n}$ on $G^{n}$ by letting
$(x_{1},...,x_{n})\sigma_{i}=(x_{1},...,x_{i}x_{i+1}x_{i}^{-1},x_{i},x_{i+2},...,x_{n})$. I wonder what algorithms ...
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Characterizations of subgroups of $S_n$ with small centralizer
Are there nice characterizations/classifications of subgroups of the symmetric group $S_n$ whose centralizers (in $S_n$) are of a particular type?
In particular, I'm interested in the case where the ...
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An almost Right Angled Artin Group?
Let $G$ be the group defined by the following presentation.
$$G=\left\langle v,w,x,y,z | [v,w],[w,x],[x,y],[y,z],[z,v],vw^{-1}xy^{-1}zv^{-1}wx^{-1}yz^{-1}\right\rangle.$$
Is it true that the subgroup ...
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Inner representation and related group $C^*$-algebra
Let $G$ - discrete group. Consider $C^*$-algebra $C^*_\gamma(G)\subset B(\ell^2(G))$ which is generated by operators $T_g:\delta_x \mapsto \delta_{gxg^{-1}}$ where $g\in G$. Are there some good ...
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Cayley nomenclature
Suppose I have a finitely generated group $G$ acting on a set $X,$ and we make a graph, whose vertex set is $X,$ and where we join $x_1$ to $x_2$ by an edge labeled $g_i$ if $g_i(x_1) = x_2.$ What is ...
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Growth of products of large sets in Lie groups
The Steinhaus Theorem states that if $A$ is a set of positive measure in a locally compact group $G$, then the set $A *A^{-1} := \{ab^{-1} | a,b \in A \}$ contains a ball around the identity in $G$.
...
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Example of a group satisfying Max which is not polycyclic by finite [duplicate]
Are there examples of a group satisfying max condition which is not polycyclic by finite. I was looking into Finiteness conditions-I by D J S Robinson, and it was posed as a question there, but the ...
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cospectral graphs have isomorphic adjacency matrices?
Let $G$ and $H$ be two cospectral graphs, then can we say that their adjacency matrices are isomorphic?
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Automorphism of a graph which is like Kneser graph
Thanks for any help or comment.
Consider the following graph:
$X=\{1,2,\dots,6\}$ and a vertex of graph is the following set $Y=\{a,b\}\cup\{c,d\}$, where $a,b,c,d\in X$. There exist an edge between ...
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Proof/reference for a variant of Pontryagin duality
Let $X,X'$ be locally compact abelian groups with a non-degenerate quadratic form
$\left<\bullet ,\bullet \right>\colon X\times X' \to \mu_{l}$,
where $l$ is a prime, and $\mu_l$ the group of $...
- 3,362
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Automorphism group of $(\Bbb Z\times\Bbb Z) \rtimes \Bbb Z$
Automorphism group of $\Bbb Z\times \Bbb Z \times \Bbb Z$ is $SL(3,\Bbb Z)$. I am wondering what the automorphism group of $(\Bbb Z\times \Bbb Z) \rtimes \Bbb Z$ would be. Generator c of $\Bbb Z$ is $...
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is the stabilizer of a point in a Mobius group virtually abelian
Currently I am reading a paper "Infinite group actions on spheres" by Gaven Martin. I am a first year graduate students and I got lots of questions, so one of them is about the following example: (...
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(When) can the presentation in Steinberg's Yale notes fail to give an algebraic group?
I'm trying to understand a remark which appears on p. 1483 of Cohen, Murray and Taylor's "Computing in Groups of Lie Type." It says, "We have not used the presentations described in [7] or [30] ...
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Laplacian on two Lie groups have the same Lie algebra
I know that if $G$ is a Lie group and $\mathfrak g = span\{X_i, 1\leq i \leq n\}$ be its Lie algebra, where $\{X_i\}$ are the vector fields of $G$. Then, the Casimir-Laplacian of $G$ is given by
$$\...
- 650
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Generalizing approximate $\mathbb{Z}$-equivariance of a simple function
Let $f(x) := x^2 + (1-x^2)x$ and $F(x) := \log \frac{x}{1-x}-\frac{1}{x}$. It can be shown (cf. https://math.stackexchange.com/questions/1865370/) that $F$ is approximately equivariant w/r/t the $\...
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Induced structure of topological group [closed]
If we consider a closed Jordan curve $\mathcal{C}$, I know that it's homeomorphic to the circle $S^1$. Now I take an homeomorphism $\phi:S^1\longrightarrow\mathcal{C}$ and this homeomorphism induces a ...
- 1,252
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The number of fixed points of an automorphism of $\mathbb{Z}_m\times\mathbb{Z}_n$
Let $m$ and $n$ be two positive integers such that the groups $\mathbb{Z}_m$ and $\mathbb{Z}_n$ have no common direct factor. Then an automorphism $f$ of $\mathbb{Z}_m\times\mathbb{Z}_n$ is of type
$$\...
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Computing subgraph orbits
I have group $G$ acting on a 4-regular 120 node graph $\Gamma$. I want to compute equivalence classes of connected subgraphs of $\Gamma$, where by equivalent I mean in the same orbit. More ...
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When does a normal subgroup H of a group G have a complement in G? [closed]
When does all normal subgroups of a group have complement? This question is different from question When does a subgroup H of a group G have a complement in G?
Related to this question I ask is ...
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Isomorphism with fixed number of Permutations [closed]
Suppose, I have a fixed number of permutations for each sub-graph to determine isomorphism of whole graph. Is it possible to determine efficiently ?
For example, $G, H$ are isomorphic graphs. For ...
- 267
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A non-surjective coboundary map induced by a central extension
Let $k$ be a number field and
$$ 1\to A \to B \to C \to 1$$ be a central extension of finite groups over $\mathcal{O}_k$ (the ring of integers of $k$), with $B$ non-commutative. Consider the induced ...
- 11
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261
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Presentation of hyperbolic groups [closed]
Is it true that all hyperbolic groups are finitely presented? If yes, what is the right reference for that?
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Show that $\mathrm{SL}_2(\mathbb{F}_p)$ is quasi-random
Terry Tao gives this oblique definition of quasirandom group in his notes 3
$G$ is quasi-random (of order $D$) if all non-trivial unitary representations $\rho: G \to U(H)$ have dimension at least $...
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Is $n+\frac {1}{2}$ in Kendall-Mann numbers and quantum harmonic oscillator related?
It is known that quantum harmonic oscillator is connected to the symmetric group of infinite order which is isomorphic to the permutation group. According to Cayley's theorem any finite group is ...
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An optimal lower bound related to generators in a boolean interval of finite groups
Let $[H,G]$ be a rank $n$ boolean interval of finite groups (i.e. $[H,G] \simeq B_n$ as lattice).
Let the set $E = \{ g \in G \ | \ \langle H,g \rangle = G \}$
Remark: If $g \in E$ then $Hg \...
