Questions about the branch of abstract algebra that deals with groups.
0
votes
0answers
2 views
If d(“G/H”) < d(G) = 2, must H contain a primitive element?
Let $G$ be a finite group that can be generated by $2$ elements, and let $H \leq G$ be a (not necessarily normal) subgroup for which there exists some $g \in G$ such that $H \langle g\rangle = G$. ...
7
votes
2answers
254 views
Residual finiteness: why do we care? [on hold]
Residually finite groups have been studied for a long time. However, I am struggling to work out why we care, or perhaps, why they continue to be of interest. Let me explain.
Magnus, in his 1968 ...
6
votes
0answers
118 views
Polynomial growth without Gromov's theorem
It is a consequence of the theorem of M. Gromov on groups with polynomial growth and a result of P. Pansu(*) that if a finitely generated group $\Gamma = \langle S\rangle,\, S=S^{-1}$ satisfies ...
1
vote
1answer
48 views
An example of a non-(locally cyclic) Abelian group whose automorphism group is cyclic not of order $2$
In the wake of my curiosity on this kind of things, I was thinking if there is an example of a non-(locally cyclic) Abelian group whose automorphism group is cyclic not of order $2$. Every example I ...
-1
votes
0answers
61 views
Represention of the element of a group: A Confusion [on hold]
I am reading a paper "FAST ALGORITHMS FOR CALCULATION OF GIBBS DERIVATIVES ON FINITE GROUPS" by R. S. Stankovic (Approx. Theory & its AppL 7:2, June 1991). In Section 2, following is written about ...
8
votes
3answers
185 views
How do small central extensions drop the dimension of a faithful representation?
Apologies in advance that this is a very soft question. I might be talking complete nonsense. But I hope I am talking about something that has even been studied...
I am interested in the phenomenon ...
2
votes
1answer
82 views
Involution on the components of a group algebra
If $G$ is a finite group and $k$ a field, there is a canonical involution (ie an involutive anti-automorphism) $\sigma$ on $k[G]$ induced by $g\mapsto g^{-1}$. Given that the center of $k[G]$ has ...
0
votes
0answers
62 views
On a property of split short exact sequences [migrated]
Let $A_{\bullet}, B_\bullet$ and $C_\bullet$ be three short exact sequences of groups (not necessarily abelian) out of which $A_\bullet$ and $B_\bullet$ are split. Assume that there is again a short ...
-3
votes
0answers
69 views
Proalgebraic completion [on hold]
For a finitely generated group, say Γ, what is the meant by of the proalgebraic completion of Γ? I came across this while seeing a paper on Representation Growth for Linear Groups by Larsen and ...
10
votes
1answer
407 views
A group whose automorphism group is cyclic
Is there an Abelian group $A$ which is not locally cyclic whose automorphism group is cyclic ?
This question was first posted here.
13
votes
1answer
300 views
Number of solutions to equations in finite groups
Suppose $G$ is a finite group and that $E$ is an equation of the form $x_1 x_2 ... x_n = e$, where each $x_i$ is in the set of symbols $\{x, y, x^{-1}, y^{-1}\}$.
Is it always true that the number ...
1
vote
1answer
181 views
Why a tensor product of $2\times 2$ unitaries cannot implement a $3\times 3$ unitary?
Let $\{v_1, \dotsc, v_m\} \in \mathbb{C}^{2^n}$ be a set of orthonormal vectors. Define a map $R_m$ from $2^n \times 2^n$ to $m \times m$ matrices as follows:
$$R_m(M) := \sum_{i,j=1}^m (v_i^*M v_j) ...
1
vote
0answers
81 views
Generalization of groups with non-connected prime graph
Suppose $G$ is a non-abelian finite group and $\pi(G)=\pi_1(G)\cup \pi_2(G)$ is a disjoint union of all primes dividing $|G|$. Suppose further that for every two elements $g_1,g_2\in G\setminus Z(G)$ ...
12
votes
0answers
246 views
Linear groups which don't contain products of free groups
Let $G \subset GL(n, \Bbb Z)$ be a f.g. linear group. The Tits alternative says that $G$ is either virtually solvable (i.e. has a solvable subgroup of finite index), or contains a free group $F_2$. ...
4
votes
0answers
90 views
Improvements of the Reidemeister-Schreier index formula for particular classes of groups
I have a couple of questions regarding possible improvements of the Reidemeister-Schreier index formula: let $G$ be a $d$-generated group and let $H$ be a subgroup of $G$, then
$$d(H) \le (d-1) ...
0
votes
1answer
147 views
Decomposing a reducible representation of the unitary group
Consider the representation $L_U$ of the unitary group $U(n)$ on $L(\mathbb{C}^n)$ where $L_U$: $L(\mathbb{C}^n) \rightarrow L(\mathbb{C}^n)$ is a linear operator that $L_U M=U M U^{\dagger} $, ...
6
votes
1answer
142 views
When are toral orbits in buildings the difference of fixed-sets?
Let $L$ be a $p$-adic field, let $G$ be a reductive group over $L$ (I'm even okay assuming semisimplicity for now). Let $T$ be a maximal torus of $G$. Let $B$ be the building for $G(L)$. (Edit 1: ...
4
votes
1answer
231 views
Can an abelian group have a minimal group topology?
In the abstract of this paper, it is said that a minimal group topology on an abelian group is not Hausdorff.
Suppose $G$ is an abelian group and $\mathcal T$ is a minimal group topology on $G$ and ...
0
votes
0answers
76 views
$\text{Hom}(G,\mathbb{Z})$ [duplicate]
Fix a cardinal $\kappa$ and consider $\mathbb{Z}^\kappa$ with componentwise addition and the subgroup $$F_\kappa :=\{g:\kappa \to \mathbb{Z}: \{\alpha\in \kappa: g(\alpha)\neq 0\} \text{ is ...
5
votes
1answer
151 views
English translation of Witt's paper on the Mathieu groups?
Does anyone know of an English (or French) translation of Witt's paper Die 5-fach transitiven Gruppen von Mathieu ? (It's the one in which the Witt design is introduced. Well, I guess.)
Here's an ...
1
vote
0answers
59 views
Quotient groups of “Abelian-times-compact”, what are they called?
In what I am doing now this class of groups appears all the way: (Hausdorff) quotient groups of $A\times K$, where $A$ are locally compact abelian groups, and $K$ compact groups.
I wonder, if this ...
8
votes
1answer
189 views
Embedding the group von Neumann algebra into an injective von Neumann algebra on the same Hilbert space
Let $\Gamma$ be a discrete group, $\newcommand{\VN}{\rm VN}$
and let $\VN(\Gamma)$ denote its von Neumann algebra, regarded as a subalgebra of ${\sf B}(\ell^2(\Gamma))$. It is well known that ...
2
votes
1answer
185 views
Structure of symplectic group over finite fields
We are working over the finite field $\mathbb{F}_{q}$ of odd prime characteristic $p$ and of cardinality $q$ some power of $p$. We recall the symplectic group $Sp(4,\mathbb{F}_{q})$ as the group of ...
17
votes
3answers
552 views
Center of a simply-connected simple compact Lie group and McKay correspondence
Let $G$ be a simply-connected simple compact Lie group. Its center $Z(G)$ is a finite abelian group, say $Z(G) = \mathbb Z/k\mathbb Z$ for $G=SU(k)$.
I find the following interpretation of $Z(G)$ in ...
1
vote
2answers
106 views
Is the poset of all precompact group topologies on an abelian group $G$, order-isomorphic to $\operatorname{Sub}(\hat{G})$?
In this page, in abstract, it is claimed that the poset of all Hausdorff precompact group topologies on an abelian group $G$, is order-isomorphic to the the subgroup lattice of $\hat{G}$, the ...
3
votes
0answers
73 views
When do two lattices have the same stabilizer in the diagonal torus?
This is moved from MSE, where I asked and didn't receive an answer (see http://math.stackexchange.com/questions/1145151/lattices-in-mathbbq-pn-with-the-same-stabilizer)
Let $T$ be the diagonal torus ...
0
votes
2answers
203 views
Continuous families of finite subgroups of a Lie group
Suppose we have a continuous family of finite subgroups of a compact Lie group G. All the subgroups are necessarily isomorphic. Alternately, we can say we have a continuous family of homomorphisms ...
3
votes
0answers
76 views
Is there a Noetherian profinite group of infinite rank?
Is there a profinite group $G$ such that any closed subgroup $H \leq G$ is finitely generated, but there is no $n \in \mathbb{N}$ such that every closed subgroup of $G$ can be generated by at most $n$ ...
4
votes
1answer
118 views
Can any finite distributive weighted lattice be realized by inclusion of groups?
By theorem 2.1 here, any finite distributive lattice $\mathcal{L}$ can be realized as an intermediate subgroups lattice.
A weighted lattice $(\mathcal{L},\tau)$ is a lattice $\mathcal{L}$ with a ...
2
votes
0answers
93 views
Torsion in profinite groups
Is there a finitely generated profinite group $G$ with a closed subgroup of infinite index $K \leq G$ such that for every $g \in G$ there exists some $n \in \mathbb{N}$ for which $g^n \in K$ ?
Can ...
3
votes
1answer
158 views
Centralizers in the universal central extensions of the alternating groups?
For $n \ge 8$ the Schur multiplier $H_2(BA_n, \mathbb{Z})$ (where $A_n$ denotes the alternating group) stabilizes to $\mathbb{Z}_2$, and hence there is a universal central extension $\widetilde{A}_n$ ...
1
vote
0answers
87 views
Virtually abelian centralizers
This is a sort of a follow-up question to Limits of conjugated subgroups (though it might not seem at first glance to have much to do with it.)
Anyway, I'm wondering what sort of groups have the ...
8
votes
2answers
337 views
Can any finite lattice be realized as an intermediate subgroups lattice?
Let $G$ be a finite group and $H$ a subgroup.
Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$.
Question: Can any finite lattice be realized as ...
0
votes
1answer
98 views
The simple groups with an absolutely irreducible projective representations with small degrees
In the question "Simple Hurwitz Groups of order less than 10^7", it came up that all the Hurwitz groups with absolutely irreducible projective representations of degrees up to 7 over any field are ...
6
votes
1answer
114 views
Abelianization of Hilbert modular group
Let $d>0$ be a square free positive integer and let $\mathcal{O}_d$ be the ring of integers in $\mathbb{Q}[\sqrt{d}]$. What is the abelianization of the Hilbert modular group ...
2
votes
2answers
101 views
admissible characters for $PGL_{2}(F)$
What are the irreducible admissible representations of $PGL_{2}(F)$ for $F$ a local nonarchimedean field and do we have formulas for their characters?
3
votes
1answer
123 views
Condition(s) for the full autormophism group $\operatorname{Aut}(C(G, S))$ of the Cayley graph of $G$ to be isomorphic to $G$
If $\Gamma = C(G, S)$ is the (undirected) Cayley graph of a finite group $G$ with generating set $S$, then $G \le \operatorname{Aut}(\Gamma)$, the "full" automorphism group of $\Gamma$.
When is ...
1
vote
0answers
86 views
Intuitive meaning of benign subgroup
Disclaimer! This is a copy of a question I posted on M.SE!
I still think the question belongs there but I'm not getting any answers so I'm dublicating with slight changes:
I've been studying a proof ...
1
vote
0answers
70 views
Hyperspecial parahoric group schemes/Chevalley groups
Let $G$ be a simple group over $k=\mathbb{C}$, $A=k[[t]]$, $K=k((t))$, and consider the group $G(K)$. This group is a split reductive group over a local field, and therefore the results of Bruhat and ...
4
votes
2answers
211 views
Geometrisation of inclusion-like epimorphisms to free groups
Let $H_g$ be the standard $3$-dimensional handle-body, whose boundary is denoted $S_g$, the oriented closed surface of genus $g\geq 1$.
Call $F_g$ be the free group of rank $g$.
Denote by $i:S_g \to ...
6
votes
0answers
169 views
Groups $G$ with a subgroup $H$ of finite index isomorphic to $G$
I am looking for simple examples of finitely generated residually finite group $G$ with a subgroup of finite index $H<G$ isomorphic to $G$. There are virtually nilpotent groups with this property, ...
-2
votes
0answers
14 views
Finite index of automorphisms that fix normal subgroup [migrated]
does anyone have some thoughts on this:
Let G be a group and K a normal subgroup in G of finite index. Then the set X of all automorphisms phi of G that fix K (meaning phi(K)=K, so not necessarily ...
3
votes
2answers
300 views
symmetric 2-cocycle / many projective representations
Let $G$ be a finite group, $k$ the field of complex numbers.
Are there (cohomologically nontrivial) group 2-cocycles $\sigma\in Z^2(G,k^\times)$ such that for all $g,h\in G$:
...
5
votes
1answer
341 views
Simple Hurwitz Groups of order less than 10^7
I'm trying to calculate a table of all simple hurwitz groups of order less than 10^7. None of the tables I found went further than 10^6, so I decided to use the tables of all simple groups up to 10^7 ...
8
votes
1answer
152 views
Is the Cayley graph of Thompson's group isolated in the space of vertex-transitive graphs?
Consider Thompson's group (the one commonly referred to as $T$), which is finitely presentable. Consider the Cayley graph, but then forget the coloring and direction on edges. So now we just have an ...
5
votes
0answers
148 views
Derived subgroup of rational points versus rational points of derived subgroup
Let $\mathbf G$ be a connected algebraic group defined over a field $\mathbb F_p$. If $q=p^n$, then the groups $\mathbf G^\prime (\mathbb F_q)$ and $\mathbf G (\mathbb F_q)^\prime$ are not always ...
9
votes
1answer
180 views
Asymptotic dimension of $C'(1/6)$ small cancellation groups
Do there exist finitely presented $C'(1/6)$ small cancellation groups with arbitrarily high asymptotic dimension?
To offer a little more motivation, Roe proves that all hyperbolic groups have finite ...
1
vote
1answer
97 views
p-groups and 2-generated abelian images
Let $p$ be a prime number. Is there a finite nonabelian $p$-group $G$ such that any finite epimorphic $2$-generated image of $G$ is abelian?
2
votes
0answers
246 views
A dual version of a theorem of Øystein Ore in group theory
This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved:
Theorem: Let $(H \subset G)$ be an inclusion of finite groups such that the lattice ...
5
votes
0answers
175 views
Conjugation of the quotient of $SL(n,\mathbb{C})$ by a finite subgroup
EDITED Let $G={SL}_{n,{\mathbb{C}}}$, the special linear group over ${\mathbb{C}}$.
Let $H\subset G$ be a finite subgroup.
Set $X=G/H$ be the corresponding homogeneous space, it is a complex variety.
...
