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Lower bound for restricted sumset in ordered groups

Recently in The restricted sumsets in finite abelian groups it is proved that Suppose that $k \geq 2$ and $A$ is a non-empty subset of a finite abelian group $G$ with $|G| > 1$. Then the ...
navashree chanania's user avatar
4 votes
1 answer
378 views

Where to begin in Computational Group Theory?

I'm coding a small application that looks for periodic solutions to the gravitational n-body problem. I'm trying to better understanding the symmetries of solutions, which is made up of the product of ...
G. Fougeron's user avatar
0 votes
0 answers
61 views

Defining rank of an abelian subgroup using the second centralizer

I recently posted this on MSE, but didn't receive any feedback; so I'm posting it on MO. I recently came across this article which explored the maximal abelian subgroups of the symmetric group $S_n$. ...
dbossaller's user avatar
9 votes
1 answer
304 views

About the normal subgroups of Burnside groups

I was reading "On periodic groups of odd period $n\ge 1003$" of V. S. Atabekyan. He found that the Burnside group $B_n$ with $n\ge 1003$ has uncountably many normal subgroups. However, I was ...
GroupKing's user avatar
2 votes
0 answers
167 views

Centralizer of PSL in PGL and of SL in GL: reference request

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\PSL{PSL}$Consider the general linear group $\GL(n,q)$ over a finite field with $q$ elements and ...
Nick Belane's user avatar
9 votes
1 answer
1k views

Is the number of varieties of groups still unknown?

A variety of groups is a class of groups satisfying a specified set of equations. Equivalently, it is a class of groups that is closed under homomorphic images, subgroups, and direct products. A ...
Martin Brandenburg's user avatar
4 votes
0 answers
97 views

Characterization of Vilenkin group

It is shown in [1, Section 1] by C.W. Onneweer that every infinite compact, metrizable, zero-dimensional commutative group is a Vilenkin group. My question is does this implication also hold if we ...
John's user avatar
  • 85
12 votes
1 answer
816 views

Do linear groups over a commutative ring satisfy the Tits alternative?

A group $G$ is said to satisfy the Tits alternative if any finitely generated subgroup of $G$ is either virtually solvable or contains a nonabelian free subgroup. Tits proved this for linear groups ...
Nobody's user avatar
  • 863
8 votes
1 answer
321 views

Does every cancellative duo semigroup embed into a group?

Prompted by the comments to a recent answer by YCor to a related question (here), I'd like to ask the following: Q. Does every cancellative duo semigroup embed into a group? A (multiplicatively ...
Salvo Tringali's user avatar
1 vote
0 answers
89 views

The base group of a wreath product of an abelian group by $ {\mathbb{Z}}$ is a characterstic subgroup

I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can direct me to some relevant results. Let $A$ be a finitely generated abelian group,...
ghc1997's user avatar
  • 823
9 votes
1 answer
223 views

$p$-adic analytic pro-$p$ group satisfies a pro-$p$ identity?

Let $p$ be a prime. Let $w$ be an element of a free pro-$p$ group $F_r$ of finite rank $r\geq 2$. Then we say that a pro-$p$ group $G$ satisfies the pro-$p$ identity $w$ if for every homomorphism $ f:...
stupid boy's user avatar
8 votes
2 answers
596 views

If a semigroup embeds into a group, then is it a subdirect product of groups?

The title has it all: Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups? If yes, then $S$ is a subdirect product of subdirectly irreducible groups,...
Salvo Tringali's user avatar
2 votes
1 answer
232 views

Reference for surjectivity of the canonical map $R^{G_1} \otimes R^{G_2} \to R^{G_1 \cap G_2}$

Let $R$ be a commutative ring, $G$ a finite group with an action over $R$. Let $G_1, G_2 \subset G$ be two subgroups. Then the canonical map $R^{G_1} \otimes R^{G_2} \to R^{G_1 \cap G_2}$ is ...
Zheming Xu's user avatar
3 votes
0 answers
161 views

Generalized dimension property for rings

My question is very basic, I am looking for a characterization (and name) of rings $R$ satisfying the following property $\star$. For any $V, W$ two finitely generated $R$-modules such that $V\oplus W\...
GSM's user avatar
  • 223
1 vote
0 answers
172 views

Isomorphism classes of finite $\mathbb{N}$-groups

Where can I find resources on isomorphism classes of finite $\mathbb{N}$-groups, i.e. groups acted on by the monoid $(\mathbb{N}, +)$? I edited this question to be more focused on what I'm interested ...
Keith's user avatar
  • 591
1 vote
1 answer
209 views

A question about automorphism group of abelian group

Does anyone know any references that describe automorphism group $\operatorname{Aut}(\mathbb R^n\times \mathbb T^m)$? I searched for a long time but couldn't find it.
free's user avatar
  • 71
7 votes
1 answer
224 views

Generating set of permutation group such that generators do not "contain" other group elements

Let $(G, X)$ be a permutation group with domain $X$. Let $O=\{o_1,\dots,o_m\}$ be the set of orbits of $G$. I am interested in generating sets $S$ with the following property: Let $g\in S$ be a ...
Martin Rubey's user avatar
  • 5,822
5 votes
0 answers
140 views

Classification of visible actions for *reducible* representations?

Is there a classification of the pairs $(G,V)$ such that $G$ is reductive [and connected, if you like], and $G$ acts faithfully and visibly on $V$ - crucially, including all cases where $V$ is ...
Joshua Grochow's user avatar
11 votes
1 answer
250 views

Recognising the elements of the Grigorchuk group

The Grigorchuk group $\mathfrak{G}= \langle a,b,c,d \rangle$ is a group of automorphisms of the infinite rooted binary tree $\mathcal{T}_2$. Every element of $\mathfrak{G}$ can be represented by a ...
AGenevois's user avatar
  • 8,401
1 vote
0 answers
92 views

The existence of such homomorphism [closed]

Are there any papers or books that investigate/discuss the relationship between conjugacy classes and normality for the existence of non-trivial homomrphism f:G->H were H is some nontrivial ...
Naif's user avatar
  • 61
2 votes
0 answers
118 views

What are the finite-dimensional irreducible unitary representations of $E(3)$?

Let $E(3)$ be the Euclidean group of $\mathbb{R}^3$ defined, e.g., by $$E(3)=SO(3)\ltimes T(3)$$ where $T(3)$ is the translation group. I am looking for a reference classifying all the finite-...
PontyMython's user avatar
2 votes
1 answer
111 views

Structure of elements of a finite group not contained in any conjugate of a proper subgroup

Let $G$ be a finite group and $H$ be a proper subgroup of $G$. It is elementary to prove that the union of all conjugates of $H$ under $G$, $$U:=\bigcup_{\sigma\in G}\sigma^{-1}H\sigma,$$ is properly ...
Nicolas Banks's user avatar
4 votes
0 answers
75 views

Alternating bihomomorphism is a skew 2-cocycle

It seems to be a well-known fact that every alternating bihomomorphism $G\times G\to\mathbb{C}^\times$ for a finite abelian group $G$ is the skew of some 2-cocycle (see for instance Symmetric analogue ...
Josep's user avatar
  • 41
5 votes
0 answers
108 views

Non-monotileable amenable groups

This is crossposted from MSE. We say a subset $A$ of a group $G$ is a monotile for $G$ if $G$ is a disjoint union of right translates of $A$. In his article Monotileable Amenable Groups, B. Weiss ...
Saúl RM's user avatar
  • 10.6k
4 votes
0 answers
107 views

Complex reflection groups: reference request

Suppose that $V$ is a finite-dimensional complex vector space, that $m\ge 2$ is an integer and that $G\subset \operatorname{GL}(V)$ is a finite subgroup such that $V$ is an irreducible ${\mathbb{C}}[G]...
inkspot's user avatar
  • 3,137
2 votes
0 answers
68 views

Amplification argument for hyperlinear groups

Let us define a group $G$ to be a hyperlinear group if it satisfies the conclusion of Theorem 3.6. in these notes by Vladimir Pestov. It is well-known that one can use the so-called amplification ...
Keivan Karai's user avatar
  • 6,224
6 votes
0 answers
138 views

Equation in a nilpotent group

Let $G$ be a nilpotent group of class at most $r$ (that is, $\gamma^{r+1}G=1$). Let elements $g_1,\dotsc,g_n\in G$ be fixed. We are interested in the set $V\subseteq\mathbb Z^n$ of solutions $x=(x_1,\...
Semen Podkorytov's user avatar
16 votes
3 answers
1k views

Conjectures in the representation theory of the symmetric group

Question: What are current open conjectures about the representation theory of the symmetric group? I am interested mostly in the characteristic 0 case, but conjectures for the modular case can also ...
Mare's user avatar
  • 26.5k
3 votes
1 answer
165 views

When the fundamental group of subgraph of groups embeds?

Given a connected graph of groups $\mathcal G$ (where edge maps are embeddings), by a subgraph we mean a graph of groups obtain by omitting some vertices, some edges, and replacing the remaining ...
tomasz's user avatar
  • 1,338
3 votes
0 answers
186 views

Bourgain-Gamburd-like theorems in the non-algebraic case

For $\mu$ a Borel probability measure on the compact group $G=\operatorname{SU}(d)$, Bourgain-Gamburd prove that the spectral radius of the associated operator on $L^2(G)$ is strictly less than one, ...
John Rached's user avatar
1 vote
0 answers
71 views

Finitely presentable groups are residually finite if and only if they are universally pseudofinite

Suppose $G$ is finitely presentable. Then residual finiteness of $G$ is equivalent to $G$ satisfying the universal theory of finite groups (equivalently, to every existential statement true in $G$ ...
tomasz's user avatar
  • 1,338
1 vote
0 answers
134 views

Isomorphic quotients of a countably infinitely-generated free abelian group

Let $F$ denote the free abelian group on countably infinite generators. I am trying to understand the relationship between normal subgroups $A$ and $B$ of $F$ with isomorphic quotients. So is there a ...
medvjed's user avatar
  • 11
6 votes
0 answers
121 views

Sylow subgroups of the restricted Burnside group $\mathrm{RB}(d,n)$?

$\DeclareMathOperator\RB{RB}$What is known about the Sylow subgroups of the restricted Burnside groups $\RB(d,n)$ ? I am looking for a reference. In fact my question is slightly more general. Recall ...
user521482's user avatar
5 votes
0 answers
200 views

Virtual fibring of $\mathrm{Out}(F_2\times F_2)$

A finitely generated group $G$ is said to virtually fibre if there is a finite index subgroup $H\leq G$ and a non-trivial map $\varphi:H\to\mathbb{Z}$ with $\ker(\varphi)$ finitely generated. I want ...
Marcos's user avatar
  • 911
5 votes
1 answer
365 views

Number of $k$-tuples of elements generating a cyclic group

Let $k$, $m$ be natural numbers, and $C_m:=\mathbb{Z}/ m \mathbb{Z}$ be the cyclic group of order $m$. Let $N_{k, \, m}$ be the cardinality of the following set: $$\{(a_1, \ldots, a_k) \in (C_m)^k \; ...
Francesco Polizzi's user avatar
1 vote
0 answers
188 views

Does every amenable group $G$ admit a two-sided Folner sequence?

By two-sided Følner sequence I mean a sequence $(F_N)_N$ of subsets of $G$ which is both a left-Følner and a right-Følner sequence. Context: I just came up with this question and surprisingly I haven'...
Saúl RM's user avatar
  • 10.6k
1 vote
0 answers
110 views

Character table of $\mathrm{P\Gamma L}_2(q)$ with $q$ even

Let $q = 2^f$ for some integer $f\geqslant 3$. The character table of $\mathrm{SL}_2(q)\cong\mathrm{PSL}_2(q)$ can be deduced from the character table of $\mathrm{GL}_2(q)$ (see, for example, Exercise ...
Groups's user avatar
  • 379
1 vote
1 answer
232 views

Transfer for the group of coinvariants: a reference request

Let $G$ be a group and $M$ be a $G$-module, that is, an abelian group written additively on which $G$ acts: $$ (g,m)\mapsto g m.$$ We consider the group of coinvariants $$ M_G:=G/\langle g m -m\ |\ g\...
Mikhail Borovoi's user avatar
9 votes
0 answers
254 views

An identity for characters of the symmetric group

I am looking for a reference for the identity $$\chi_\lambda(C)=\frac{\dim(V_\lambda)}{|C|}\sum_{p\in P_\lambda,\,q\in Q_\lambda,\,pq\in C}\operatorname{sgn}(q)$$ for the irreducible characters of the ...
Hjalmar Rosengren's user avatar
17 votes
1 answer
1k views

Explicit character tables of non-existent finite simple groups

In connection with the historical development of the classification of finite simple groups, I am interested in a particular aspect that seems to be less well-documented than the main narrative of ...
Sebastien Palcoux's user avatar
2 votes
0 answers
106 views

Hereditarily just-infinite pro-$2$ groups

An infinite profinite group $G$ is called just-infinite if all non-trivial closed normal subgroups of $G$ have finite index. A profinite group is called hereditarily just-infinite if every open ...
stupid boy's user avatar
2 votes
0 answers
137 views

$p$-adic Banach group algebra

Let $G$ be a discrete group. Consider the Banach $\mathbb{Z}_p$-algebra: $$c_0(G, \mathbb{Z}_p) = \{ F : G \to \mathbb{Z}_p \mid \lim_{g \to \infty} |F(g)|_p = 0 \}$$ with the product given by the ...
Luiz Felipe Garcia's user avatar
6 votes
1 answer
285 views

Classification of algebraic groups of the types $^1\! A_{n-1}$ and $^2\! A_{n-1}$

This seemingly elementary question was asked in Mathematics StackExchange.com: https://math.stackexchange.com/q/4779592/37763. It got upvotes, but no answers or comments, and so I ask it here. Let $G$ ...
Mikhail Borovoi's user avatar
1 vote
1 answer
214 views

Conditions of P for existence of orthogonal matrix Q and permutation matrix U satisfying QP = PU

Question: Let $P\in \mathbb{R}^{d\times n}$ be a $d$-rank real matrix and $PP^T = c I_d$ with a certain constant $c > 0$. Under what additional conditions of $P$ does there exist an orthogonal ...
Eddie's user avatar
  • 187
9 votes
1 answer
735 views

Where has this structure been observed?

$\newcommand{\M}{\mathcal{M}}$Let $M$ be a monoid. Consider the following structure: $R_X,R_Y:\mathbb{Z}^2 \to M$ satisfying the following "compatiblity-relation": $$R_X (x, y) \cdot R_Y (x +...
Asaf Shachar's user avatar
  • 6,741
2 votes
0 answers
88 views

Simple modules and trivial source modules

Let $(K,\mathcal{O},k)$ be a large enough $p$-modular system. In this context, I would like to ask what is known about the following question: when are simple $kG$-modules trivial source modules? So ...
Bernhard Boehmler's user avatar
16 votes
1 answer
408 views

Is there a relationship between Broué's abelian defect group conjecture and Alperin's weight conjecture?

Let $G$ be a finite group, let $k$ be a large enough field of characteristic $p>0$. Let $p\mid |G|$. Broué's abelian defect group conjecture states the following: Let $B$ be a block of $kG$ with ...
Bernhard Boehmler's user avatar
1 vote
0 answers
83 views

$p$-modular splitting systems and the characteristic of the ring $\mathcal{O}$

Let $k=\overline{k}$ be a field of characteristic $p$. Let $(K,\mathcal{O},k)$ be a $p$-modular system. Let both $k$ and $K$ be splitting fields for $G$ and its subgroups. The ring $\mathcal{O}$ can ...
Stein Chen's user avatar
6 votes
2 answers
366 views

Twisted forms with real points of a real Grassmannian

Let $X={\rm Gr}_{n,k,{\Bbb R}}$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb R}^n$. We regard $X$ as an ${\Bbb R}$-variety with the set of complex points $X({\Bbb C})={\rm Gr}_{n,k,{\...
Mikhail Borovoi's user avatar
15 votes
2 answers
613 views

Existence of a regular semisimple element over $\mathbb{F}_{q}$

This is probably old, a Chevalley level of old, but I'm not at all an expert in this field so I need help. Let $G$ be a simply connected (almost) simple linear algebraic group defined over $K=\mathbb{...
D. Dona's user avatar
  • 455

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