All Questions
Tagged with reference-request gr.group-theory
700 questions
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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 ...
4
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1
answer
378
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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 ...
- 297
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0
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61
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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 $S_n$. ...
- 195
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1
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304
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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 ...
- 91
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167
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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 ...
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1
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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 ...
- 63.1k
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97
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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 ...
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816
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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 ...
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322
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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 ...
- 10.5k
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89
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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,...
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$p$-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:...
- 667
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596
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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,...
- 10.5k
2
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1
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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 ...
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161
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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\...
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172
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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 ...
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1
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209
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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.
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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 ...
- 5,822
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140
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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 ...
- 3,230
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250
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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 ...
- 8,401
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92
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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 ...
- 61
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118
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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-...
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1
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111
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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 ...
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75
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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 ...
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108
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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 ...
- 10.6k
4
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107
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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]...
- 3,137
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68
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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 ...
- 6,224
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138
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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,\...
- 809
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3
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1k
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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 ...
- 26.5k
3
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1
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165
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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 ...
- 1,338
3
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186
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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, ...
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71
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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 $G$ ...
- 1,338
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134
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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 ...
- 11
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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 ...
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5
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200
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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 ...
- 911
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1
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365
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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: $$\{(a_1, \ldots, a_k) \in (C_m)^k \; ...
- 66.3k
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0
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188
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Does every amenable group $G$ admit a two-sided Folner sequence?
By two-sided Følner sequence I mean a sequence $(F_N)_N$ of subsets of $G$ which is both a left-Følner and a right-Følner sequence.
Context: I just came up with this question and surprisingly I haven'...
- 10.6k
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110
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Character table of $\mathrm{P\Gamma L}_2(q)$ with $q$ even
Let $q = 2^f$ for some integer $f\geqslant 3$. The character table of $\mathrm{SL}_2(q)\cong\mathrm{PSL}_2(q)$ can be deduced from the character table of $\mathrm{GL}_2(q)$ (see, for example, Exercise ...
- 379
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Transfer for the group of coinvariants: a reference request
Let $G$ be a group and $M$ be a $G$-module,
that is, an abelian group written additively on which $G$ acts:
$$ (g,m)\mapsto g m.$$
We consider the group of coinvariants
$$ M_G:=G/\langle g m -m\ |\ g\...
- 14.1k
9
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254
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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 ...
- 2,306
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1
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1k
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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 ...
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answers
106
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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 ...
- 667
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137
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$p$-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 ...
- 1,485
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1
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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 $G$ ...
- 14.1k
1
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1
answer
214
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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 ...
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735
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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 +...
- 6,741
2
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0
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88
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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 ...
- 1,755
16
votes
1
answer
408
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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
...
- 1,755
1
vote
0
answers
83
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$p$-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 ...
- 311
6
votes
2
answers
366
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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,{\...
- 14.1k
15
votes
2
answers
613
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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{...
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