Questions about the branch of abstract algebra that deals with groups.

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1
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0answers
123 views

Does SL(3,q) have a subgroup of order $q^3.(q^3-1)$ [closed]

Let $q=p^n$ for $p>3$.I want to know whether the group $G_2(q)$ has a subgroup of order $q^3.(q^3-1)$. First I look the paper of Peter Kleidman. In this paper Kleidman determined all maximal ...
2
votes
1answer
166 views

What is the growth of the rank of a power of a finite simple group?

Which asymptotic bounds (upper and lower) are known for $s_n$ - the minimal number of generators of $S^n$ where $S$ is a nonabelian finite simple group?
10
votes
0answers
213 views

Solving a set of equations in a finite symmetric group

A standard way to find solutions to a finite set of equations in a finite symmetric group ${\rm S}_n$ is to take the equations as relators of a finitely presented group, to use the low index subgroups ...
2
votes
1answer
160 views

About direct limit of groups

Let $G_i$ be sequence of groups for $i\in \mathbb N$ and Let $\phi_i$ be a monomorphism from $G_i$ to $G_{i+1}$. Let $\Sigma$ be the direcet limits of $G_i$ under the embeddings of $\phi_i$. Let ...
3
votes
0answers
160 views

Alternating quotients of (2,3,7;10)

It was shown that the only quotient of the group $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{10} \rangle$ of the form PSL(2, q) is PSL(2, 41). That lead me to consider other families of simple ...
4
votes
1answer
95 views

Finiteness properties for graph of groups decompositions

My curiosity was raised by the following question and the huge variety of comments and suggestions it attracted. I wondered if a converse statement might be equally interesting. Let $G$ be a finitely ...
3
votes
2answers
249 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 $H \langle g\rangle = G$. ...
18
votes
6answers
1k views

Residual finiteness: why do we care?

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
176 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
63 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 ...
8
votes
3answers
233 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
96 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 ...
10
votes
1answer
462 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.
14
votes
1answer
347 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 ...
2
votes
1answer
204 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
94 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 $g_1,g_2\in G\setminus Z(G)$ ...
13
votes
0answers
271 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
94 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) ...
2
votes
1answer
241 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
210 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
242 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
83 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
160 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
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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
232 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
216 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
650 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
3answers
156 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 ...
5
votes
0answers
170 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
243 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
82 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
121 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
103 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
209 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
104 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
354 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
106 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
132 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
107 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
152 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
92 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
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0answers
89 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
226 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
193 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, ...
3
votes
2answers
324 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
350 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
168 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
164 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
196 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
102 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?