Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,182 questions
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The Frattini subgroup of $D_{\infty}$ [closed]
Please hint me. $\phi(D_{\infty})?$ $\phi(G)$ is Frattini subgroup of $G$, intersection of all the maximal subgroup of $G$ and
$D_{\infty}=<x,y|x^2=y^2=1>$.
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Missing formula! [closed]
I am doing a project on group association schemes, in particular looking at the structure constant
$$p_{KL}^M = \#\{(x, y, xy) : x \in K, y\in L, xy \in M\}$$
where $K, L$ and $M$ are conjugacy ...
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The automorphisms of a 2-group of nilpotency class 2
Let $p$ be a Merssene prime, i.e. $p=2^a-1$, where $a$ is a prime.
Let $R$ be a 2-group of order $2(p+1)=2^{a+1}$. Also we know that $|Z(R)|=2$ and $R/Z(R)$ is abelian.
Can we conclude that $R$ has ...
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The number of cyclic subgroups
Let $G$ be a finite non-cyclic group such that it has cyclic subgroup of order $n$. Please consider the following claim:
The number of cyclic subgroups of order $n$ in $G$ is a multiple of the ...
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The maximal Abelian subgroups of PSL(3,q).2 the extension of PSL(3,q) by the graph automorphismAutomorphism
Hello
For some part of my research I want to know that PSL(3,q).2, the extension of PSL(3,q) by the graph automorphism, has the same maximal abelian subgroups as PSL(3,q)?
Also if f is a field ...
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Name for a particular subgroup of parabolic subgroups of the general linear groups. [duplicate]
Possible Duplicate:
Name for a particular subgroup of parabolic subgroups of the general linear groups.
Let $V$ be vector space. The subgroup $P$ of $GL(V)$ consisting of all automorphisms ...
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The symmetry group of $\mathbb Z^d$
Let $d \ge 1$, and consider the integer lattice $\mathbb Z^d$. This is a homogeneous space, in the manner of the Erlangan Programm.
I would like to write $\mathbb Z^d = G / H$, where $G$ is the ...
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Positive function with zero Haar integral
If $C$ is a compact (semi-)topological (semi-)group, are there nonzero positive functions having zero Haar integral? In other words: is the Hermitian product associated to the Haar integral degenerate?...
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(co)homology of cyclic groups
Hello to all,
While sprucing up my knowledge of group (co)homology,I stumbled onto the following question: The first step you usually take to compute various (co)homologies is to construct the ...
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Transitive map on a profinite group
Let $f$ be a continuous endomorphism of a compact Hausdorff totally disconnected topological group $G$ and let $H$ be a closed normal subgroup of G such that $f(H)\subseteq H$ and with $\mu(H)=0$ ...
- 361
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Fixed points free automorphisms of Teichmüller spaces
Let $\mathcal{T}_{g,n}$ be the Teichmüller space of a compact oriented surface of genus $g$ with $n$ marked points. Assume that $N:=3g-3+n>0$. Viewing $\mathcal{T}_{g,n}$ as a bounded domain in $\...
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Conjugacy of Cartan subgroups in $\mathrm{GL}(n)$
$\DeclareMathOperator\SL{SL}$I have probably a very basic question on the structure of semisimple Lie groups. Sorry if it is too elementary.
Let either $G=\SL(n,\mathbb{R})$ or $G=\SL(n,\mathbb{C})$. ...
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Trying to solve for the remainder of $a^q$ modulo $q$
Let $q$ be a prime and $a$ be a number from $0$ to $q-1$ (not an equivalence class).
The elements $a^q$ are exactly the elements of order $q-1$ modulo $q^2$.
I'm trying to solve the equation:
$$a+2*\...
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Examples of isomorphic non-equivalent twisted group algebras
Let $F$ be a field, $G$ be a finite group and $\alpha \in Z^2(G, F^*)$ . The twisted group algebra $F^{\alpha}G$ is a $F$-algebra with $F$ vector basis, $\{\bar g : g \in G \},$ and multiplication ...
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Ramifications in Galois closures of number fields
Let $K/\mathbb{Q}$ be a finite Galois extension, and $L/K$ an infinite Galois extension such that the Galois group $ \operatorname{Gal}(L/K) $ is isomorphic to a closed subgroup of $ \operatorname{GL}...
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The center of Sylow subgroups
$\DeclareMathOperator\Syl{Syl}$Let $G$ be a finite group, $\Phi(G)$ is the Frattini subgroup of $G$. And $G/\Phi(G)$ is a simple group. Let $P\in \Syl_{p}(\Phi(G))$, where $p\in \pi(\Phi(G))$ and $\...
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The intersection of product of Sylow subgroups
Suppose $G$ is solvable, and $\pi(G)= \{2,m,n\}$, $O_{2}(G)=1$. Then can we use the solvability of $G$ to prove that $O_{2^{\prime}}(G) \neq 1$? Let $\bar{G}= G / O_{2^{\prime}}(G)$, what about $O_{...
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Examples of non-proper profinite HNN extensions
We define a profinite HNN extension as the profinite completion of the abstract HNN extension. In the abstract case, the homomorphim of the base group to the HNN extension is always a monomorphism. ...
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624
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Irreducible representations of finite p-groups
Let $G$ be a finite $p$-group. What are irreducible representations of $G$ over a field of characteristic $q$, such that $(p,q)=1$ ? Can we say something in general ? In particular, if there exists ...
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Isomorphism between two groups [closed]
Let $\mathbf{F}_q$ be finite field of order $q$, where $q$ is an odd-prime (or power of an odd prime). Is there an isomorphism between the following subgroups of unipotent $3\times 3$ matrices over $\...
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Probability distribution of random products of elements of a generating set of a finite non-abelian group
Let $G$ be a finite non-abelian group, and consider a choice of $N$ distinct elements $g_{0},g_{1},\ldots,g_{N-1}\in G$ that generate $G$. Now, let $t$ be an arbitrary positive integer, and let $d_{1},...
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A subset (or subgroup) associated to a group
Edit: According to comment conversations we revise the question.
Let $G$ be a group. We consider the following subset of $G$:
$$\{g\in G \mid e^{\lambda_g} \in \mathbb{C}\lambda (G)\},$$
where $\...
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Number of cycles under a certain action on Z/nZ [closed]
Computer scientist here looking at a question that came about from in-place matrix transposition, but rusty on my abstract algebra and number theory...
Suppose we have the multiplicative group $\...
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Confusion on translating k-fold transitivity of groups from Endliche Gruppen by Huppert
The definition 1.7 from Endliche Gruppen, B.Huppert, vol-I, Chap.II, Pg.148 is as follows: Die Permutationsgruppe $\mathfrak G$ auf der Ziffernmenge $\Omega$ heißt $k$-fach transitiv $(k \leq |\Omega|...
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When is the set of $n$-th powers in a group a subgroup? [closed]
Let $G$ be a non abelian group and $G_n=\{x^n | x\in G\}$ and n is integer. Is there a sufficient condition that makes $G_n$ be a subgroup of $G$ for arbitrary $n$?
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number of orbits of a proper subgroup
Let $G$ be a permutation group that acts on (say) $X=\{1,2,..,n\}$, and $H$ be a proper subgroup of $G$. Can one say anything precise about when the number of orbits of $H$ on $X$ will be equal to ...
- 133
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Existence of a cyclic non-normal subgroup in a $p$-group
Let $G$ be a finite non-abelian $p$-group, where $p$ is an odd prime,
$N$ be a normal subgroup of $G$ of order $p$, where $\frac{G}{N}$ is non-abelian.
Does there exist an element $g\in G$ such that ...
- 487
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Do we have a one to one correspondence between positive roots and reflections in a Coxeter group?
By the answer of the question, the set of reflections of a Coxeter system $(W,S)$ is given by $R = \{ wsw^{-1} : w \in W, s \in S\}$.
Do we have a one to one correspondence between positive roots and ...
- 6,201
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Number of Boolean algebra subintervals in weak order of $S_n$
I'm wondering if anybody has an easy way to compute the number of subintervals in weak order of $S_n$ (considered as a Coxeter group of type $A_{n-1}$) that are isomorphic to Boolean algebras. I know $...
- 2,168
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A question on permutations
Given integers $1$ through $n$, let $S$ be set of ordering of integers that respect even alternating or reverse alternating permutations (https://en.wikipedia.org/wiki/Alternating_permutation) up to ...
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SO(3) transformation that produces a reflection [closed]
This came up doing some research in quantum information. Let us consider two orthogonal three-dimensional unit vectors $v$ and $w$
$v^T\cdot w=0$,
and the Householder transformation
$H=I_{3}-2v\...
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on lifting extensions
Let $G$ be a connected reductive group with $G_{der}$ simply connected and $T$ a maximal torus over an algebraically field $k$.
We consider a extension $\tilde{T}$ of the maximal torus $T$ by a torus ...
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256
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Questions on invariant operators of finite group representations
1) Is there an equivalent of the Casimir operator for an irreducible representation of a finite group?
2) Given an invariant operator of a certain group, can I check if it is invariant under only ...
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Where can I find the classification of groups of order 16p? [closed]
I need to classify the groups of order $16p$ by their generators and relations between the generators. Can I find this classification anywhere?
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Projective characters with corresponding factor set
The following is just a follow up to my previous question. I have a finite group $H$ with 14 ordinary characters. The Schur multiplier $M(H)\cong 2^2$. Hence the group $H$ will have 3 sets of ...
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Products of maximal inclusions of finite groups with a non-obvious intermediate
Let $(H_1 \subset G_1)$ and $(H_2 \subset G_2)$ be core-free maximal inclusions of finite groups.
Their product, the inclusion $(H_1 \times H_2 \subset G_1 \times G_2)$, admits four obvious ...
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Isomorphism theorem for subfactors?
It's about the existence of a generalization of the first isomorphism theorem for groups, for subfactors :
Let $(N \subset M)$ and $(N' \subset M')$ be irreducible inclusions of hyperfinite $II_1$ ...
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the structure of a 2-group [closed]
Let $ M $ be a finite group of order $2^{a+1} $ and let $ M $ have a normal subgroup $ R $ such that $|M: R|=2 $. Also we know that $ R $ is an elementary abelian subgroup of $ M $.
For example $ ...
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resources in surjunctive groups
Are there any free available resources on surjunctive groups which are available to say: a graduate level student?
A textbook would be fine also.
Regards.
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question about twisted group of Lie type A_n
Let $G=PSU_3(q)$ and $q=p^n$, where $n$ is odd. Can we conclude that $PSU_3(p)$ is a subgroup of $G$?
- 391
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definition of a weakly doubly transitive group action
I'm reading Francis M. Choucroun, "Analyse harmonique des groupes d'automorphismes d'arbres de Bruhat-Tits", Mémoires de la S. M. F., tome 58 (1994), p. 1 - 166, and he speaks of a weakly doubly ...
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Torsion version of HNN extensions.
I am thinking of a version of HNN extensions as follows:
Assume $H,K$ are subgroups of a group $G$ and $\phi:H\to K$ is an isomorphism. We define
$ G_{\phi,n}$ to be the group generated by $G$ and $...
user23860
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2-sylow subgroups
Dear All, by the paper of Carter and Fong, we know the structure of 2-sylow subgroups of $GL(n,q)$. Let $W_{r-1}=Z_2\wr Z_2\wr...Z_2$ (r-1 times), and $W$ is a 2-sylow subgroup of $GL(2,q)$, then $W_r=...
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Lattice-ordered group of rational rank 1
Does there exist a lattice-ordered, not totally ordered, group of rational rank $1$?
Rational rank 1 means isomorphic to a nonzero subgroup of $\mathbb{Q}$. There exist totally ordered groups of ...
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A question about minimal simple group
I want to know a proof of this fact: "every simple group has a minimal simple group as a subquotient." (If $H$ and $K$ are two subgroups of $G$ s.t. $H\lhd K$, then $\frac{K}{H}$ is called a ...
- 487
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How to prove this equation [closed]
For any compact abelian group $K$.$$K\cong H_0\times \mathrm{U}(1)^k,$$where $H_0$ is a finite group.
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What is a "non-splitting covering" of a finite group?
Apologies if this is elementary, but I have never heard the terminology before:
What is a "non-splitting covering" of a finite group?
I encountered the term while reading this paper, in which ...
- 1,173
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809
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Injectivity of a group homomorphism with domain a free product
Hello!
Let $K = G\ast_U H$ be the free product of the groups $G$ and $H$ amalgamated along the common subgroup $U$. I have a homomorphism of groups $f:K\to K'$ defined using the universal property of ...
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Picard-Vessiot Extension over a Differential Field?
Given a differential field F and a linear algebraic group G over the constant field C of F, find a Picard-Vessiot extension of E of F with G(E/F)=G:
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intersections of $SO_2^n, SL_2^n$ with $SO_{2n}, Sp_{2n}$
Let $\mathbb{R}^{2n}$ be endowed with the canonical symplectic structure $(\omega, J)$, where $\omega$ the usual nondegenerate symplectic form and $J$ the usual almost complex structure (ie. $\omega(\...
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