Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
2,036
questions with no upvoted or accepted answers
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Soluble subgroups of 8-dimensional orthogonal groups over GF(4) transitive on nondegenerate 1-subspaces
Let $V$ be an $8$-dimensional vector space over $GF(4)$ equipped with a nondegenerate plus type quadratic form, $G$ be an almost simple group with socle $L=\Omega^+(V)$, and $H$ be a soluble subgroup ...
- 767
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140
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Central frattini extensions
Let denote by $V = V(c, s, t)$ the class of all finite $p$-groups such that $class(G) \leq c$, $expZ(G) \leq p^s$, and $G/Z(G)$ has a maximal normal abelian subgroup of rank $\leq t$.
The class $V$ ...
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69
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Geometric effects of removing elements of D2n generalizable?
So, if I start with a full Dihedral group D2n to represent a regular, ideal polygon in the hyperbolic plane, then I remove an element (and any subsequently necessary elements so that it is still a ...
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391
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Intertwining number
Hello,
I am wondering about a statement on this page.
Especially about this part:
If $\pi_1$ and $\pi_2$ are irreducible and finite dimensional or unitary, then the intertwining number $c(\pi_1,\...
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393
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Finitely presented group and its subgroups
Suppose I have a finitely presented group $G$. By this, I mean I know explicitly what $S$ and $R$ are such that $G = \langle S \mid R \rangle$. Suppose I have a subgroup generated by a finite set of ...
- 1,201
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74
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Number of subgroups of finite abelian p-groups with a certain cotype.
Given a finite abelian $p$-group $G$ of rank $r$ I'm looking for the number of elements in a group $H$ with $\mathrm{rk}(H)=r$, such that $H/\langle y\rangle\cong G$.
- 23
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195
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Quotients of Abelian Groups
Let $G$ be an abelian group and let $A$ and $B$ be subgroups of $G$. Furthermore, let $C$ be a subgroup of $A \cap B$. I would like to find another subgroup $A+B \subseteq D \subseteq G$ so that $D/(...
- 205
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about measuring lemma
I wonder if you explain that why Chermak and Delgado named it in their article by measuring lemma? Is the measuring lemma caused the measure on finite groups?
A. Chermak, A. L. Delgado, A measuring ...
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38
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Minimum number of solutions in a system of equalities and non-equalities
Let $k<N$ and $P_1, ..., P_{2k+1}, \lambda_1, ..., \lambda_k$ be elements of a finite group $G$ of size $N$.
Find the minimum number of solution of the system
$$P_{2i} + P_{2i+1} = \lambda_i, \...
- 43
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254
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minuscule representations and classical groups
Let $G$ a semisimple group over an algebraically closed field $k$.
We assume that $G$ is classical.
We call a $z$-extension, a group $\tilde{G}$ such that $\tilde{G}$ is a central extension of $G$ by ...
- 3,364
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187
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on trivialisation of T-torsors
Let $X$ a smooth connected projective curve over an algebraically closed field $k$ and $F$ its function field. $T$ a $X$-torus.
Let $R$ be any ring.
Let $E$ a $T$-torsor on $(X-x)\times_{k}R$. Does ...
- 3,364
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252
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Is $SL(2,5)$ irreducible?
Maybe this is just a very fundamental problem, but I am not too sure the answer. It is well-known that $SL(2,5)$ is contained in $SL(2,q)$ iff $q$ is odd and $5\mid q(q^2-1)$. My question is whether ...
- 133
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75
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Cubic Honeycomb Traverse
Starting at an arbitray cube in an infinite cubic lattice is there a path that visits every cube in every n^3 region in n^3 steps? I have found 2 or three such traverses for 2^3, but I have yet to ...
- 1
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67
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How to distinguish projective involutions?
Let $K$ be a field of characteristic not $2$ and $R$ a continuous von Neumann regular ring with centre $Z=Z(R)$ isomorphic to $K$. For an example one may assume $R$ is a matrix ring of $n\times n$-...
- 804
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128
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character of a finite group where characteristic of field F not equal to 0.
Why is it that if characteristic of the field F is not equal to 0 then the constant function 0 is an F-character of a finite group G?
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representations of the Lorentz group in 4 dimensions
Hi,
First of all I should say I am quite uneducated in group theory, so my question can be very naive. Sorry about that.
I'm reading Srednicki's "Quantum Field Theory" and I have a bit of trouble ...
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133
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On X-s-permutable subgroups of a finite group
I want to prove Lemma 2.1(1) in the paper On X-s-Permutable Subgroups of a Finite Group by Min Bang SU, Yang Ming LI. It is on the web.
This is my proof.
.
Since $H$ is $X−s−$permutable in $G$, then ...
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580
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Kunneth formula (finite group cohomology) for non trivial action of group
Is there a Kunneth formula relating $H^i(k[G],A)\otimes_k H^i(k[G],B)$ and $H^i(k[G],A\otimes_k B)$ where $A\otimes_k B$ is given the diagonal $G$ action ?
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582
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128
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A kind of orthogonal subgroup
Let $n$ a positive integer and $k \in \mathbb{Z}^n$ such that for all integer $a \geq 2$ and $h \in \mathbb{Z}^n$ we have $k \neq ah$. Here $\cdot$ is the scalar product.
Is it true that $\{x \in \...
- 255
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278
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Modular representations of the symplectic group
Let G=Sp(2m,2) be a finite symplectic group acting on $F_2^{2m}$. This group G acts 2-transitively on $\Omega_{+}$ and on $\Omega_{-}$. Let $F$ be an algebraic closure of $F_2$.
I am interested to ...
- 2,139
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301
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Automorphism group of algebraic function fields
Let $K$ be a finite field and let $F/K$ be a function field. Is it possible to deduce the genus of $F/K$ from the automorphism group of $G=Aut(F/K)$?
Is it possible to do so if we know that $|G|$ is ...
- 2,139
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178
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semigroup actions of groups on regular rooted trees
If $G$ is a group which has a semigroup action on a regular rooted tree via prefix-preserving, continuous transformations (I give the tree the path metric), what kinds of algebraic restrictions can we ...
- 125
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176
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Group theoretical properties and symmetry based representation of common functions
Dear All,
I'm searching for some references, whether someone already studied the group theoretical properties of functions. There are some very basic symmetries, like parity, but is there a set of ...
- 1
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355
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self-centralizers in p-groups
Let $p$ be a prime number and $P$ a $p$-group.
(1) If $A$ is a maximal Abelian subgroup, what are nice examples where it isn't self-centralizing?
(2) What if $A$ happens to be normal as well?
- 1,160
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167
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The derived representation and $H_\pi^\infty$
There is some (probably stupid) thing that I did not get in Serge Lang's $SL_2(\mathbf{R})$: On page 93 he considers a representation $\pi:G\to GL(H)$ of a group $G$ in a Banach space. Then he ...
- 12.2k
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682
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Unitary and injective representation of a free group
Is there an example of an injective homomorphism
$\pi: F_2\to U(n)$ of the 2 generator free group $F_2$
in some unitary group of matrices $U(n)$?
- 155
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263
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What is the bottom of a directed-complete partial order of groups?
Given a directed-complete partial order of finitely presented groups, I want to say that the free group is the bottom, but I don't think that is right.
Can anyone say what is a typical bottom in a ...
- 1,227
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171
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$R^2 \cdot s_1$ is a central extension of $R^2$ by $s_1$
Hi,
I am currently working with Jacobi groups and I am not clear why $R^2 \cdot s_1$ is a central extension of $R^2$ by $s_1$
$R^2 \cdot s_1$ is the following constructed group:
The elements of $R^...
- 13
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200
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Finite index in $F_m$ implies non-trivial intersection
I was looking at a paper, and I saw this claim,
It is obvious that if $H$ has finite index in $F_m$ then $H$ has non-trivial intersection with each of the non-trivial subgroups of $F_m$.
Why is ...
- 1
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161
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Subtori of restriction of scalars
Let $E/F$ be a finite separable extension of a commutative field $F$. Let $T$ be the torus
${\rm Res}_{E/F}\; {\mathbb G}_m$, where ${\rm Res}$ is Weil's restriction of scalar. Is there a simple ...
- 5,966
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372
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Amenability of an "almost Hamiltonian" group
Here is another interesting question that I can't answer on my own.
Let $G$ be a countable, discrete group such that for any subgroup $H$ of $G$ and any element $s$ of $G$ we have $[H : sHt]$ is ...
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1
answer
195
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Perfect Cayley graphs for abelian groups have $\frac{n}{\omega}$ disjoint maximal cliques
Let $G$ be a perfect/ weakly perfect Cayley graph on an abelian group with respect to a symmetric generating set. In addition let the clique number be $\omega$ which divides the order of graph $n$. ...
- 1,841
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130
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Isomorphism of Stable groups formed by different sequences
I have been thinking the relation between two stable groups formed as follow:
1) $GL(2,\mathbb R)\subseteq GL(3,\mathbb R)\subseteq\cdots GL(n,\mathbb R)$,
$G(\infty,\mathbb R)=\cup_{n=2}^{\infty} GL(...
-2
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1
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181
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What do you call continous transformations that preserve the finite group structure?
A number of years ago I studied a preon model (Journal of Mathematical Physics 38:3414-3426, 1997) in which the preons interacted like group elements. I thought it curious that you could sometimes ...
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1
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207
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class structure constants relation
Let $C_{j,k}^l$ ,usually called class structure constants, eg Jansen and Boon and/or JQ Chen, be the number of times the class $l$ is generated from the product of classes $j,k$ and $c_j=c_{-j}$ (a ...