Questions about the branch of abstract algebra that deals with groups.
-2
votes
1answer
70 views
on the solvable groups of order $p^aq^b$
We know that if $ p$ is a prime number then $ O^p (G) $ is the smallest normal subgroup of $ G $ such that $ G/O^p (G) $ is a $ p $-group.
Now let $ G $ be a finite group of order $ p^aq^b $ where $ ...
-2
votes
0answers
40 views
a question about fixed-point-free automorphisms 1 [on hold]
let G be a finite group with a fixed-point-free automorphism a of order 3. Prove that [x,y,y]=1 for all x,y in G.
0
votes
0answers
38 views
Finite Subgroups Of $GL(2,\mathbb{R})$ [migrated]
I have the following question:
Is it true that every finite subgroup of odd order in $GL(2,\mathbb{R})$ is cyclic?
Thanks!
2
votes
2answers
114 views
Group of homomorphisms with real coefficients and circle coefficients
Do you know some groups $G$ that have the same group of homomorphisms to $R$ and to $S^1$; i.e. $$Hom(G,R) = Hom(G,S^1)??$$ Is there any special property for $G$ in order to satisfy the last relation?
...
2
votes
0answers
136 views
The tallest possible lattice?
Let O be a complete discrete valuation ring and G a finite group. Recall that a finitely generated O-free OG-module $M$ such that the traces of the invertible endomorphisms of $M$ generate a strictly ...
3
votes
0answers
75 views
Groups such that all finite-dim representations are finitely presented
Let $G$ be an infinite group. What sorts of finiteness properties can I put on $G$ to ensure the following holds for all $M$?
Let $M$ be a finite-dimensional vector space over $\mathbb{Q}$ upon ...
15
votes
1answer
258 views
Is there a natural notion of completion of a Coxeter system?
Let $(W,S)$ be a Coxeter system. Then any element of $W$ can be written as a finite products of elements of $S$. I want some notion of a "completion" of $W$, call it $\hat{W}$, whose elements are ...
9
votes
0answers
237 views
Does anybody know if the Fourier algebra of SL(3,Z) has an approximate identity?
(Note to those who like to tidy LaTeX, or ${\rm \LaTeX}$: I kindly request that you don't put any LaTeX in the title of this question, nor change the bolds below to blackboard ...
4
votes
1answer
133 views
Which hyperbolic tilings are Cayley graphs?
I realise the question is easy but after asking to a few people (and never getting a clear answer), I thought it could be instructive to ask it here:
Given a regular tiling of the hyperbolic plane is ...
-5
votes
0answers
64 views
A question about the inverse of a matrix [closed]
and
it's obvious that det A =w=1 or -1.
My question is simple, but it seems to be chaos.
I will be very happy if someone can replay.
1
vote
0answers
123 views
Centralizers of elements in HNN extensions
Let $G$ be a (countable) group, $H<G$ be a proper subgroup and $\theta:H\to G$ be an injective group homomorphism such that $\theta(H)\neq G$. Consider the HNN extension $\Gamma=<G,t \mid ...
5
votes
1answer
292 views
Finite groups for which the element orders form an arithmetic progression
Which are the finite groups $G$ such that the element orders of $G$ form an arithmetic progression? Several remarks:
$S_3$, $A_4$ and any $p$-group of exponent $p$ satisfy this property.
If $G$ ...
6
votes
1answer
247 views
Using math software to show that the following groups are infinite?
I would like to show that the following finitely presented group in 3 generators $P, Q, R$ is infinite in certain cases:
$$P^p, Q^q, R^r, (PQ)^2, (QR)^2, (PQR)^2, (QR^{r/2+1})^a (RQR^{r/2})^b$$
For ...
2
votes
1answer
103 views
Proving that a closed walk of some odd length k exists on a Cayley graph
I'm trying to prove the following:
Let $G$ be a group with finite symmetric generating set $S$ and let $\Gamma(G,S)$ be the corresponding Cayley graph. Let $X_1, X_2,\cdots$ be a simple random walk ...
4
votes
0answers
141 views
More on Moonshine for the Thompson group and weakly holomorphic weight one half modular forms
This question is a follow-up to
Monstrous Moonshine for Thompson group $Th$?
and is based on various comments to that question, in particular S. Carnahan's mention of the
connection to known ...
0
votes
0answers
92 views
Infinite groups admitting field structure [migrated]
Does every infinitely generated Abelian group admit a field structure?
(i.e. If $(G,+)$ is an infinitely generated Abelian group, then,
is there a binary operation "$\cdot$" such that $(G,+,\cdot)$ is ...
0
votes
0answers
154 views
A second isomorphism theorem for the inclusions of groups
The usual second isomorphism theorem for groups is: let $G$ be a group, $S$ and $N$ subgroups with $N$ normal, then $SN$ is a subgroup of $G$, $S\cap N$ is a normal subgroup of $S$ and $SN/N \simeq ...
-3
votes
0answers
87 views
Telescopic groups and inverse Galois problem [closed]
I define the notion of "telescopic group" in http://math.stackexchange.com/questions/695576/telescopic-groups
Is any telescopic group isomorphic to the Galois group of an extension of the rationals?
...
3
votes
1answer
73 views
Siegel domains and cuspidal functions
Let $F$ be a number field and $\mathbb{A}$ the ring of adeles over $F$. We consider $P_{n}$ the mirabolic subroup of $GL_{n}$.
Do we have a analog of Siegel subset for the quotient ...
7
votes
2answers
422 views
What is the name of this type of groups?
Suppose $A$ is a finite set and $\Sigma=A\cup A^{-1}$. Let $L\subseteq \Sigma^{\ast}$ be a regular language on the alphabet $\Sigma$. Is there a common name for the group $G$ presented as:
$$G=\langle ...
13
votes
2answers
545 views
Groups which are only defined up to conjugation
I'm trying to understand what the right way is to think about "groups which are only well-defined up to conjugation." Since this is somewhat vague let me clarify it by pointing out the main examples ...
4
votes
1answer
137 views
Can Haar measure fail to be bi-invariant without conjugation shrinking a set?
(This is a slightly reformatted and clarified version of my question from math.SE, since I believe
the answer there is wrong and its poster has not responded to my comment in over two weeks.)
Let ...
3
votes
0answers
154 views
Groups whose finite index subgroups are isomorphic
I am looking for examples of the following situation:
$G$ is an infinite group. Every two finite index proper subgroups of $G$ are isomorphic.
The only examples that I have now are (1) ...
2
votes
1answer
198 views
on the prime divisors of $(p^2+1)/2 $
The following question is equivalent to a problem in group theory.
Let $ p > 13$ be a prime number distinct from 239. Let $ a=(p^2+1)/2 $. Is there any prime divisor $r$ of $a$ such that $r\mid ...
4
votes
1answer
231 views
Locally finite compact groups
I assume all tolpological groups here to be Hausdorff. A group is called locally finite if every finitely generated subgroup is finite. What can be said about a locally finite compact group? Must it ...
3
votes
1answer
153 views
Normal intermediate subgroup and normal core
Let $G$ be a finite group and $H$ a subgroup.
The normal core of $H$ in $G$ is $core_G(H) := \bigcap_{g \in G}g^{-1}Hg$
Definition: $K$ is a normal intermediate subgroup of the inclusion $(H ...
1
vote
0answers
70 views
Real analytic diffeomorphisms of sphere
We know that for $n\ge 4$, $SL(n,\mathbb Z)$ acts real analytic diffeomorphism on the sphere $S^2$ through a finite image, what about $SL(3,\mathbb Z)$? That is to say, if $\Phi: SL(3,\mathbb Z)\to ...
4
votes
0answers
69 views
Groups of operators between local unitaries and full unitaries
Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...
6
votes
2answers
173 views
Modifying Dehn's algorithm to allow equal length replacements?
I'm an analyst trying to understand a certain class of finitely presented groups (one example is below) so it's quite likely this question is naive but I hope it is at least intelligible. Given a ...
5
votes
1answer
178 views
Kaplansky's idempotent conjecture for Thompson's group F
Let $K$ be a field and $G$ be a torsion-free group. Kaplansky's idempotent conjecture states that the group ring $K[G]$ does not contain any non-trivial idempotent, i.e. if $x^2=x$ then $x=0$ or ...
0
votes
1answer
68 views
Is a weakly separable group always Lindelöf?
By "weakly separable" I mean the notion for uniform spaces used by David Wigner and Lawrence Brown: a uniform space is weakly separable if any uniform cover has a countable subcover. For a topological ...
-1
votes
0answers
73 views
A question about perfect groups 1
G is a finite group, it is not simple, G=G',then we call G is a perfect group.
My question is if there is a classification for perfect groups。
3
votes
0answers
205 views
On the Groups of Order $(p^2+1)/2$
A few days ago I asked a question (Groups of order $p(p^2+1)/2$) about a finite group of order $p(p^2+1)/2$ and I got a lot of useful information about it. Thanks for the nice and very helpful ...
0
votes
0answers
75 views
Normal subgroups of finite solvable groups [migrated]
Let $G$ be a finite solvable group, $N$ a nontrivial abelian normal subgroup of prime exponent $p$. Let $Q$ be a $p$-Sylow subgroup of $G$ containing $N$.
Is it possible that the normal core of ...
4
votes
1answer
161 views
Boundaries of relatively hyperbolic groups
When the interior of an n-manifold $M$ has a pinched negative curvature metric of finite volume, then its fundamental group $\Gamma=\pi_1M$ is relatively hyperbolic relative to the parabolic groups ...
6
votes
5answers
656 views
Groups of order $p(p^2+1)/2$
It seems that when $p>3$ is a prime, then each group of order $p(p^2+1)/2$ is abelian as I checked by Gap for small $p$. Is it true for each $p$?
Thanks for your answers
15
votes
1answer
433 views
Is a left topological group which is a manifold a topological group?
Let $G$ be a left topological group, i.e. a topological space with group operation such that left multiplication $L_g : x \mapsto gx$ is continuous (but right multiplication and inversion are not ...
2
votes
2answers
103 views
Normal subgroups of automorphism group of infinite cubic tree
Let $T$ denote the unique (countably infinite) tree in which every vertex has degree $3$ and let $G=Aut(T)$ be the group of graph automorphisms of $T$.
I'm interested in the properties of this ...
4
votes
4answers
543 views
Normalizer of SL_2(Z) in GL_2(R)
What is the normalizer of ${\mathrm{SL}}_2({\Bbb Z})$ in ${\mathrm{GL}}_2({\Bbb R})$? Namely
${\mathrm{N}}_{{\mathrm{GL}}_2({\Bbb R})}({\mathrm{SL}}_2({\Bbb Z}))$?
1
vote
1answer
138 views
subgroups of General linear group with two generators
In General linear group $GL(n,q)$, every matrix is conjugate with its rational canonical form. Using this fact we have the conjugacy classes of cyclic groups. My question is about the groups which are ...
21
votes
0answers
609 views
Monstrous Moonshine for Thompson group $Th$?
I. As a background, in Traces of Singular Moduli (p.2), Zagier defines the modular form of weight 3/2,
$$g(\tau) = \frac{\eta^2(\tau)}{\eta(2\tau)}\frac{E_4(4\tau)}{\eta^6(4\tau)}=\vartheta_4(\tau)\, ...
6
votes
0answers
86 views
How much do the classes of geodesic metrics and Green metrics on a Gromov hyperbolic group differ?
Background
Let $G$ be a Gromov hyperbolic group. If $G$ acts properly discontinuously and cocompactly on a proper geodesic metric space $X$, then any bijective identification of $G$ with its orbit ...
0
votes
0answers
106 views
Is the direct product of distributive inclusions of groups, modular?
Let $H$ a subgroup of $G$ and $\mathcal{L}(H \subset G)$ the lattice of intermediate subgroups ($\mathcal{L}( G)$ if $H= \{ e \}$).
Definitions: A lattice $(L, \wedge, \vee)$ is :
- Distributive if ...
0
votes
0answers
58 views
Groups and triangle-square complexes
I would like to know what kind of groups (and/or their group presentation) acting geometrically on CAT(0) curved piecewise Euclidean triangle-square complexes.
Thanks
0
votes
2answers
102 views
Products of maximal inclusions of finite groups with a non-obvious intermediate
Let $(H_1 \subset G_1)$ and $(H_2 \subset G_2)$ be core-free maximal inclusions of finite groups.
Their product, the inclusion $(H_1 \times H_2 \subset G_1 \times G_2)$, admits four obvious ...
0
votes
1answer
204 views
Centralizer of derived subgroup
In all questions suppose $G$ metabelian p-group such that
G is not regular ( so $cl(G) \geq p$ ), G is not a wreath product;
$Z(G) \leq \phi(G)$.
1) Let $M$ normal abelian subgroup of $G$ such ...
1
vote
1answer
113 views
About the power subgroups of the Bianchi groups
Let $B$ be the Bianchi group and let $B^2 =\langle x^2 | x \in B\rangle$, the power subgroup of $B$.
Is it true that $B \ne B^2$ all the time ?
1
vote
1answer
134 views
JSJ-decompositions of hyperbolic groups and elementary vertices
My question is the following:
In Bowditch's JSJ-decomposition of hyperbolic groups, can elementary (virtually-cyclic) vertices have degree 1? If not, why not?
I had thought for a long time that ...
2
votes
0answers
143 views
Existence of inclusions of finite groups with a particular lattice property
Definition : Let $\sim$ be the equivalence relation on inclusions of finite groups, generated by :
$(H \subset G) \sim (\phi(H) \subset \phi(G))$, with $ \phi: G \to L$ a finite group morphism and ...
3
votes
1answer
146 views
Permutation groups transitive on partitions into ordered pairs
The following came up in a problem on graph reconstruction. It isn't very important, but I thought some people here might find it interesting and not too trivial (I'm not a group theorist).
Take a ...
