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

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4
votes
0answers
104 views

Infinite simple p-groups with only trivial irreps in characteristic p

Is there a prime $p$ and an infinite simple $p$-group $G$ such that for any field $K$ of characteristic $p$ the only irreducible $KG$-module, whether finite or infinite dimensional, is trivial (that ...
3
votes
1answer
146 views

For a cross section $\sigma\colon G/N\to G$, how is $\sigma(y)^{-1}\sigma(x)^{-1}\sigma(xy)$ called?

Let $G$ be a locally compact group, let $N$ be a closed normal subgroup of $G$, and let $\sigma\colon G/N\to G$ be a cross section. Let us define $\alpha\colon G/N\times G/N \to N$ by the formula $$ ...
3
votes
0answers
102 views

Frequency of a representation of SO(3)

When generalizing the basic tenets of Fourier Theory to the symmetric group $S_n$, we can define a notion of the frequency of a basis function (i.e. an irreducible representation of $S_n$). In ...
4
votes
1answer
174 views

Decomposing $(\mathbb C^n)^{\otimes m}$ as a representation of $S_n\times S_m$

$V=\mathbb C^n$ is a $\mathbb CS_n$-module, where $S_n$ is the symmetric group of degree $n$, via the representation sending a permutation to the corresponding permutation matrix. The tensor power ...
3
votes
0answers
190 views

Galois correspondence subgroups/subsystems

In this paper (1998) by M. Izumi, R. Longo, S. Popa, there is the following result (page 49) on compact groups: By applying this result to finite groups, we get a Galois correspondence ...
1
vote
0answers
104 views

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

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
139 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?
8
votes
0answers
174 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
139 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 ...
0
votes
0answers
67 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 ...
-1
votes
0answers
54 views

Name and theorems of transitive/Galois groups of quadratic (even) and cubic power polynomials: cyclic group extensions [on hold]

What is the transitive group details of a polynomial where only the third power terms occur? That is ${x}^{3n}+{a}_{n−1}{x}^{3(n−1)}+\cdots+{a}_{1}{x}^{3}+{a}_{0}$. I need the basic theorems that ...
3
votes
1answer
81 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
238 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
161 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
54 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
75 views

Represention of the element of a group: A Confusion [closed]

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
213 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
91 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
73 views

Proalgebraic completion [closed]

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
427 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
324 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
186 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
83 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
259 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
91 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
207 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
192 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
233 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
77 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
153 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
197 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
188 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
565 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
113 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 ...
4
votes
0answers
80 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
206 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
77 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
119 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
94 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
163 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
92 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
339 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
100 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
115 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
104 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
126 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
88 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 ...