All Questions
Tagged with reference-request gr.group-theory
700 questions
0
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Where can I find the classification of groups of order 16p? [closed]
I need to classify the groups of order $16p$ by their generators and relations between the generators. Can I find this classification anywhere?
- 11
5
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1
answer
769
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F.p. groups where all elements of the same order are conjugate
The question I want to ask is related to the Boone-Higman conjecture (see
Embedding in f.p. simple groups for the details).
We discussed recently with Ievgen Bondarenko this conjecture and he ...
- 1,437
7
votes
1
answer
818
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Uncertainty principle for non-commutative groups
Is it true that for every group $G$ and $f\in \mathbb C[G]$ it holds that $$\dim(\mathbb C[G]*f)\mathop{supp}(f)\geq |G| ?$$
Here, $\mathbb C[G]$ is the group algebra, and by $\mathbb C[G]*f$ I ...
- 2,179
3
votes
1
answer
426
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Naturality of the transfer in group cohomology
Let $G$ be a (discrete) group and $H\le G$ a subgroup of finite index. Then there is a transfer map
$$tr\colon\thinspace H^\ast(H;M)\to H^\ast(G;M)
$$
in group cohomology, where $M$ is any $G$-module ...
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1
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1
answer
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Reference on elements of finite order in principal congruence subgroups of symplectic groups
We should start with the definition of the symplectic group for an arbitrary ring $R$.
The symplectic group $Sp(g,R)$ is the subgroup of $SL(2g,R)$ such that all elements satisfy $M=J_g^t M J_g$ with $...
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0
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168
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Abelian centralizer groups (CA-groups)
I am searching for all information about CA-groups [abelian centralizer groups] and i just found a German book [Huppert] and Nilpotent Centralizer group of Suzuki in 44 pages and Group theory book of ...
2
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0
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187
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Classification of Automorphism set of a Regular graph
Let $A$ be the adjacency matrix of an $r$-regular graph $G$ with $n$ vertices (Not complete or cycle graph) . Also, let $Aut(G)$ be the set of all its automorphisms (i.e. set of permutation matrices)....
- 267
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1
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508
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Why are all involutions conjugate in the special linear group of degree 2?
It appears to be standard that the set of non-identity involutions in $SL(2, 2^n) = PSL(2, 2^n)$ forms a single conjugacy class. What is the best reference for this?
I note that
https://math....
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8
votes
1
answer
830
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Who proved that a group of polynomial growth has growth exactly polynomial?
I need to put a reference about the classical result that a f.g. group of polynomial growth has growth which is exactly polynomial.
Talking personally with people and also here in A question about ...
- 6,034
3
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0
answers
113
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Have locally principal crossed homomorphisms been studied?
Take a (multiplicative finite) group $H$ acting on the left (by automorphisms) on an (additive finite) abelian group $A$, and recall that the abelian (additive) group of crossed homomorphisms from $H$ ...
- 11.3k
3
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1
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608
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Amenability of Thompson's group looking at a 4-manifold having it as the fundamental group
Just for curiosity I have done a quick web-search and I have seen that some people are studying manifolds with amenable fundamental group. On the other hand, any finitely presented group and then, in ...
- 6,034
8
votes
1
answer
446
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Radical of $F_p[SL(2,p)]$
Let $G=SL(2,p)$. Does anyone know what is the radical of the group algebra $F_p[G]$?
Does there exists any book/paper where it is calculated?
By radical here I mean maximal ideal I of $F_p[G]$ such ...
- 2,179
4
votes
1
answer
204
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Estimates for simple random walks in groups of intermediate growth
I'm looking for references for the rate of escape and return probability for a group of intermediate growth.
Let $0<\alpha < 1$. If the volume growth is $\succeq \mathrm{exp}(n^\alpha)$, then (...
- 4,432
2
votes
2
answers
862
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Non-split groups
I am looking for a reference with definitions on what it means for an algebraic group to be split, quasi-split, and non-split. I would like to see some examples of the different "types".
Thanks,
Tom
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1
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283
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resources in surjunctive groups
Are there any free available resources on surjunctive groups which are available to say: a graduate level student?
A textbook would be fine also.
Regards.
- 1,835
13
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1
answer
736
views
Idempotent measures on the free binary system?
Let $(S,*)$ be the free (non associative) binary system on one generator (so $S$ is just the set of terms in $*$ and $1$). There is an extension of $*$ to the space $P(S)$ of finitely additive ...
- 3,547
1
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1
answer
578
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Fundamental inequality of entropy in random walks
I'm looking for a reference for an inequality related to the "fundamental inequality" about entropy and rate of escape of random walks (on the Cayley graph of a group). Namely,
$\textbf{Question}$: ...
- 4,432
3
votes
0
answers
209
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Growth of the number of generators in hyperbolic groups
Let $G$ be an infinite hyperbolic group, and let us further assume that it is residually finite (or even LERF/GFERF) so that we have plenty of subgroups of finite index.
I would like to know if one ...
- 11.3k
5
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1
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170
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Equations and random subgroups in compact groups
EDIT: Here is a more specific question.
Let $G$ be a compact group and let $w$ be a word in $d$ variables. Then the solution set $S$ of the equation of $w=1$ is a closed subset of the product $G^d$ ...
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1
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70
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A non-surjective coboundary map induced by a central extension
Let $k$ be a number field and
$$ 1\to A \to B \to C \to 1$$ be a central extension of finite groups over $\mathcal{O}_k$ (the ring of integers of $k$), with $B$ non-commutative. Consider the induced ...
- 11
3
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1
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213
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Groups with special automorphism group
I am looking for all finite groups $G$ such that for each subgroup $H$ of $G$ and each automorphism $\sigma$ of $H$ there exists an automorphism $\psi$ of $G$ whose restriction to $H$ is $\sigma$. Is ...
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552
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Who conjectured that a transitive projective plane is Desarguesian?
The only known finite projective plane with a transitive automorphism group is the Desarguesian plane $PG(2,q)$ and it seems likely that there are no others, although this is not (quite) proved.
...
- 12.7k
3
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222
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torsion free for the 2nd cohomology group?
Let $G$ denotes an infinite coutable discrete group with Kazhdan's property (T),
My question is:
is it known that the 2nd cohomology group $H^2(G,\mathbb{Z}G)$ is torsion free?
Thanks in advance!
...
- 1,528
4
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1
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773
views
Normal subgroups of projective special linear group over a ring
What are the normal subgroups of $PSL_2(\mathbb{Z}/p^n \mathbb{Z})$?
- 1,905
1
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611
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Is the automorphism group of a homogeneous (locally finite) tree unimodular?
I have seen somewhere (that I don't remember now) that the (full) automorphism group of a k-regular tree is unimodular. I assume a k-regular tree is the same thing as the homogeneous tree of degree k (...
user23860
5
votes
1
answer
712
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Structure of abelian connected complex linear algebraic groups?
Let $G$ be an abelian connected complex linear algebraic group.
Is it true that $G$ is isomorphic to $(\mathbb{G}_m)^k\times (\mathbb{G}_a)^\ell$, where the nonnegative exponents denote repeated ...
- 53
4
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1
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255
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On the divisibility of the special linear group of degree $n$ over an algebraically closed field
Let $n$ be a positive integer, $p$ a (positive rational) prime, and $\mathbb K$ an algebraically closed field. If ${\rm char}(\mathbb K) = 0$ then ${\rm GL}_n(\mathbb K)$ is divisible (see here). But ...
- 10.5k
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2
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281
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Doubly covering an even lattice
I have read that there is a way to construct a group which is a double cover of an even lattice. The very tantalizing thing about this is that if the even lattice is chosen to be the Leech lattice, ...
- 3,645
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0
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307
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On groups satisfying a law
We say that a group $G$ satisfies a law if there exists a (nontrivial) word $w \in \mathbb{F}_n$ such that $w(g_1,\dots,g_n)=1$ for every $g_1,\dots, g_n \in G$. For example, any abelian group ...
- 2,371
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2
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1k
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description of functions of conditionally negative type on a group
Recall that a kernel conditionaly of negative type on a set $X$ is a map $\psi:X\times X\rightarrow\mathbb{R}$ with the following properties:
1) $\psi(x,x)=0$
2) $\psi(y,x)=\psi(x,y)$
3) for any ...
- 1,222
1
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223
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Reference on calculation of 2nd cohomology group
Let $G$ be a finitely generated, infinite, countable discrete nonamenable group with zero first Betti number, I.e., $H^1(G, \ell^2(G))=0$, e.g., $G=F_2\times F_2$, the product of free groups of two ...
- 1,528
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193
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On a problem of Berkovich
What is the real history of the following problem proposed by Berkovich [Y. Berkovich, Z.Janko, Groups of prime power order. Volume 2, Expositions in Mathematics, 56, Walter de Gruyter, New York, 2011]...
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3
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512
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The symmetry group of $\mathbb Z^d$
Let $d \ge 1$, and consider the integer lattice $\mathbb Z^d$. This is a homogeneous space, in the manner of the Erlangan Programm.
I would like to write $\mathbb Z^d = G / H$, where $G$ is the ...
- 8,512
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0
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192
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Thin profinite groups - nonabelian analogues of p-adic integers
Let $p$ be a prime number, $S = C_p$ a cyclic group of order $p$, $G = \mathbb{Z}_p$ the profinite additive group of $p$-adic integers. It is well known that all the closed nontrivial subgroups of $G$ ...
- 11.3k
1
vote
2
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341
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Copies of ax+b inside the AN part of an Iwasawa decomposition?
As a relative novice to the structure theory of Lie algebras and Lie groups, the following is what I can gather from reading parts of Helgason's book DG, Lie groups and symmetric spaces and Knapp's ...
- 25.8k
6
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225
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Parshin's buildings for higher local fields
What is the status of the theory of buildings for higher local fields?
I know that there are some papers of Parshin, in which he describes some examples, like $PGL_2$ and $PGL_3$ over two-...
- 17.4k
15
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0
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573
views
Relation Between Truncated Braid Groups and Regular Tilings of the Complex and Hyperbolic Plane
This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question on math.SE.
There exists a rather remarkable ...
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0
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79
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Some questions about the Lévy monoid of certain densities
Let $\bf H$ be a set, $f: \mathcal P({\bf H}) \rightharpoonup \bf R$ a partial function, and $\mathcal{D}$ the domain of $f$.
Next, denote by $\mathcal M(f)$ the set of all (total) functions $\theta: ...
- 10.5k
4
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1
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686
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Character theory of $2$-Frobenius groups.
This is a crosspost of my (slightly longer) question on MSE since I'm not getting any responses there.
Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and $F_2/F=\text{Fit}\left(G/...
- 1,173
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1
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301
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Reference for a proof of the Dehn presentation
I would like a reference for a proof that the Dehn presentation is a presentation of the fundamental group of the knot complement in $\mathbb{S}^{3} $.
- 13
4
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135
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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) \...
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1
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676
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Analysis and finitely generated groups
Dear all, this is perhaps a bit a vague question, but some references would already be very helpfull.
So let $G$ be a finitely generated group and choose some finite set of generators. This allows to ...
- 8,098
7
votes
1
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499
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Posets of cosets and contractibility
For this question let $G$ be a group, perhaps infinite, and let $H_i$ for $i\in I$ be a (finite) family of subgroups closed under taking intersections. I am interested in the coset poset $\mathcal{C}(...
- 1,680
3
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0
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102
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Localized at $p$ integral representations of finite elementary $p$-groups
Let $C_p$ be a cyclic group of prime order $p$.
Let $F=C_p^n=C_p\times\dots\times C_p$ ($n$ times).
I would like to to classify finite dimensional representations of $F$ over ${\mathbb{Z}}$.
However, ...
- 14.2k
2
votes
1
answer
241
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Subgroup structure of orthogonal groups of small dimension over finite fields
How much is known about the subgroup structure of the orthogonal groups (of dimension n<=7, say) over finite fields? Can anyone point me in the direction of a good reference? I'm aware of a book by ...
- 393
5
votes
2
answers
346
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Reference request: A theorem by S. Garrison
A theorem by S. Garrison states that if $G$ is a finite solvable group and $|cd(G)| = 4$ then $dl(G)\leq |cd(G)|$ (the Taketa inequality, which is conjectured to hold for all finite solvable groups). ...
- 2,508
2
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1
answer
397
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Counting Nearest Neighbors that Stay Nearest Neighbors after Random Rearrangements
Imagine we are making necklaces with $n$ beads, each bead is a different color from all others. Let's say we make one necklace. If we make another necklace with the same $n$ differently colored ...
- 161
1
vote
1
answer
262
views
Natural actions of quotients of automorphism groups
I've stumbled upon a construction which seems to be very much classical, and yet I found nothing definite about it so far in available sources. Let $\Lambda$ be a normal subgroup of the automorphism ...
- 303
1
vote
1
answer
353
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Decomposition of an induced representation
If there is a finite group $G$ with a cyclic normal subgroup $C_n$, one can describe the indecomposable representations of $G$ through induction. How does $Ind_{C_n}^G$ decompose? For representations ...
- 13
2
votes
0
answers
909
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List of commutator identities and equivalences
Let $G$ be a group and let $[a,b]=a^{-1}b^{-1}ab$ be the commutator of $a$ and $b$ in $G$. There are several well-known commutator identities such as
$[x, z y] = [x, y]\cdot [x, z]^y$
and
$[[x, y^{-...
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