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

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### Functoriality in the group $G$ of the domain of the Baum-Connes map

Lück claims in his preliminary book, that the left hand side of the Baum-Connes map is functorial in the group $G$. For the right hand side $K(A \rtimes G)$ this is clear for the full crossed product, ...

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33 views

### Cohomology of $\mathbb Z_4$ via the Lyndon-Hochschild-Serre spectral sequence

I'm trying to understand how to construct the Lyndon-Hochschild-Serre spectral sequence for the cohomology (with integer coefficients) of the central extension $G$ of a group $Q$ by a group $N$, given ...

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71 views

### Finite groups with trivial center [on hold]

I know the Symmetric groups, Alternating groups and Frobenius groups of order $pq$, where $p$ and $q$ are distinct prime number, have trivial center. I want to know a classification of finite groups ...

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182 views

### On the Number of Parallel Automorphism Lines

Given a group $G$, one can define the transfinite line of iterative automorphisms of $G$ to be the following chain of the groups where $G_{\alpha+1}=Aut(G_{\alpha})$ for each ordinal $\alpha$ and the ...

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**7**answers

3k views

### Paradoxical Mathematical Objects Pending for Construction [duplicate]

The possible properties and applications of some mathematical objects have been described far before their rigorous mathematical definition. Some of them even had a seemingly paradoxical description ...

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**1**answer

149 views

### If a group $G$ has decidable word problem, must it have a decidable square problem?

My question is a refinement of this one about 'efficient' construction of square elements: If the word problem for a (finitely generated, finitely presented) group is decidable, must the 'square ...

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83 views

### The generalizations of solvable groups [on hold]

Please I want to ask what are the known generalizations of the concept of solvable groups and where I can find them .

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**1**answer

170 views

### Finite group with a character having one nonzero absolute value

Let $G$ be a finite group. Assume that $\chi$ is a complex irreducible character of $G$ of degree $n\geq 2$, with the property that for each element $g\in G$ either $\chi(g)=0$ or $|\chi(g)|=n$.
...

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**1**answer

251 views

### Shafarevich's theorem about solvable groups as Galois groups

I am seeking references to any proofs of Shafarevich's theorem about solvable groups being Galois groups.

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**1**answer

644 views

### Is this conjecture strictly weaker than P=NP?

My three computability questions are related to the following group theory question (first asked by Bridson in 1996):
For which real $\alpha\ge 2$ the function $n^\alpha$ is equivalent to the Dehn ...

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**1**answer

324 views

### Question on Hall's theorem

Theorem 9.3.1 in Hall's group
theory says: Let $G$ be a solvable group and $|G|=m\cdot n$, where $%
m=p_{1}^{\alpha _{1}}\cdot \cdot \cdot p_{r}^{\alpha _{r}}$, $(m,n)=1$. Let $%
\pi =\{p_{1},...,p_{r}...

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**1**answer

84 views

### Connectedness of centralizers of semisimple Lie-algebra elements under the action of a semisimple algebraic group

Let $\newcommand{\GG}{\mathbf{G}}\newcommand{\g}{\mathfrak{g}}\GG$ be a connected semisimple algebraic group over the algebraically closed field $k=\overline{\mathbb{F}_q}$, and let $\g$ be its Lie-...

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157 views

### Fricke involution on GL(3)

Define $\Gamma_0(N)=\{\begin{pmatrix}
a&b&c\\
d&e&f\\
g&h&i
\end{pmatrix}
\in SL(3,\mathbb{Z})|g\equiv h\equiv 0(\mod N)\}$ be the $N$-level congruence subgroup on GL(3).
...

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**1**answer

117 views

### Prove that if a group $G$ is generated by all cyclic subnormal subgroups, then every cyclic subgroup is subnormal

I have already found two definitions for a Baer group.
$G$ is a Baer group if it is generated by all cyclic subnormal subgroups.
$G$ is a Baer group if every cyclic subgroup is subnormal.
I want to ...

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**1**answer

203 views

### Cayley graph properties

Consider an infinite graph that satisfies the following property: if any finite set of vertices is removed (and all the adjacent edges), then the resulting graph has only one infinite connected ...

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110 views

### about a strange property of p-groups of maximal class

I am trying to look for a finite $p$-group of maximal class of order at least $p^{2p+1}$ exponent at least $p^3$ which possibly has the following property :
If s is an element in $G-G_1$ ($G_1$ is ...

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76 views

### Quandle homomorphism does not always induces group homomorphism on inner automorphism groups of quandles

Let $X$ and $Y$ be two quandles and $f: X \rightarrow Y$ be a quandle homomorphism. Then we can define a map $\bar f: Inn(X) \rightarrow Inn(Y)$ as $\bar f(S_a)=S_{f(a)}$, where $a \in X$. Then $\bar ...

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69 views

### Existence of a certain direct summand

Let $P$ be a cyclic group of order $p^n$. Let $M$ be a $\mathbb{Z}_p[P]$-module and $M_p :=\mathbb{Q}_p\otimes M$ be the associated $\mathbb{Q}_p[P]$-module. When will $M_p$ have a direct summand ...

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337 views

### The third (co)homology group

I need to prove that the third (co)homology group of a certain finitely presented group is not finitely generated. The group is not an amalgamated product or an HNN extension, and it does not act ...

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**1**answer

87 views

### Relationship between irreducible representations of the Schur covering group and elements of $H^2(G,U(1))$

Let $G$ be a finite group and let $D(g)$ be a projective representation of $G$ i.e.
\begin{equation}
D(g) D(h) = e^{i \omega(g,h)} D(gh)
\end{equation}
These can be classified by the equivalence ...

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**2**answers

257 views

### A balanced tree-like presentation of $S_3$

Does the 6-element group $S_3$ have a finite (balanced) semigroup presentation of the form $$\langle a_1,...,a_n\mid a_1=u_1, a_2=u_2,...,a_n=u_n\rangle$$ where $u_1,u_2,...,u_n$ are semigroup words? ...

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562 views

### Show that the rational cohomology ring $H^*(M;\mathbb{Q})$ needs at least two generators

Let $M$ be a simply connected closed Riemannian manifold. How does one find a necessary condition going both ways that may be imposed on $M$ (perhaps on the curvature of $M$ and on torsion) which ...

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76 views

### Certain $p$-group with cyclic center

Let $G$ be a finite $p$-group of derived length $d$, which is not a Dedekind group.
(i.e., possesses at least one non-normal subgroup).
Let $G^{(d-1)}$ be the unique normal subgroup of $G$ of order $...

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**1**answer

97 views

### Is the function field of every congruence modular curve generated by $j,j\circ g$ for some $g\in\text{GL}_2(\mathbb{Q})^+$?

So I've heard in passing that for any congruence modular curve $X$ (over $\mathbb{C}$), there is a $g\in\text{GL}_2(\mathbb{Q})^+$ such that $X$ is birational to a plane curve in $\mathbb{C}^2$ given ...

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**1**answer

819 views

### The Tall Tale of Terminating Transfinite Towers

The transfinite tower of iterative automorphisms of a group $G$ is simply definied to be the following chain of the groups where $G_{\alpha+1}=Aut(G_{\alpha})$ for each ordinal $\alpha$ and the direct ...

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137 views

### A connected Borel subgroup of the plane

It is known that the complex plane $\mathbb C$ contain dense connected (additive) subgroups with dense complement but each dense path-connected subgroup of $\mathbb C$ necessarily coincides with $\...

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**1**answer

133 views

### When is a sequence of group extensions associative?

Suppose I have groups $A,B$ and $C$ for which the following information is given:
1) The group $G_{AB}$ is a central extension of $B$ by $A$, where the abelian group $B$ acts trivially
2) The group $...

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102 views

### Small modules over finite group with large cohomology

Looking at this Example of group cohomology not annihilated by exponent of $G$? I stumbled upon one question I couldn't solve (probably because it's hard), so I post it here.
Using Lyndon resolvent, ...

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171 views

### Conjugacy class representatives for the automorphism group of a finite abelian group

Given a finite abelian group $A$, I'd like a list of conjugacy class representatives for its automorphism group ${\rm Aut}(A)$.
In fact, it's not important that I have exactly one representative from ...

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107 views

### Parallel transport for variety over finite field

I was wondering: Given a variety over a finite field, say the projective plane or sphere over $\mathbb{F}_q$. Then I can try to define parallel transport along (geodesic) curves. In particular, I can ...

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53 views

### Adjoint orbits of a finite group of type $G_2$ [reference request]

Let $q=p^\alpha$ be a prime power and $k=\mathbb{F}_q$. Let $G\subseteq \mathrm{GL}_N(k)$ be a simple finite group of Lie type, with root system of type $G_2$, and let $\mathfrak{g}\subseteq \mathfrak{...

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375 views

### primary decomposition for nonabelian cohomology of finite groups

Let $G$ be a finite group, and let $M$ be a group on which $G$ acts (via a homomorphism $G\to \operatorname{Aut}(M)$).
If $M$ is abelian, hence a $\mathbb{Z}G$-module, there is a primary ...

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**1**answer

144 views

### Subgroup lattice isomorphic to the power set lattice

If $G$ is a group, we denote by $\text{Sub}(G)$ the lattice of all subgroups of $G$, ordered by $\subseteq$. Given a cardinal $\kappa$, is there a group $G$ with $\text{Sub}(G) \cong {\cal P}(\kappa)$ ...

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140 views

### Amenable Thompson-like groups

Question: Do there exist amenable Thompson-like groups?
I realise that my question is vague, but defining and studying groups which look like Thompson's groups $F$, $T$ and $V$ seems to be an ...

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146 views

### A robust version of Schur's lemma?

Does a robust version of Schur's lemma exist? Specifically, I was wondering about something like this:
Let $B$ be a bounded operator over a vector space $V$, with underlying field $\mathbb{C}$ and ...

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99 views

### Distance regular Cayley graphs on $Z_2^n$?

Let $Z_2^n$ be group $Z_2 \times Z_2 \times \cdots \times Z_2$ with operation Exclusive-or. I'd like to know if the $Cay(Z_2^n,S)$ for $S \subset Z_2^n \setminus \{0\}$ is distance regular graph or ...

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**1**answer

136 views

### About the product of finite subsets of a torsion free group

Let $G$ be a torsion free group with identity $e$. For a subset $X$ of $G$, denote by $X^\#$ the set $X\setminus\{e\}$. Let $A$ be a finite subset of $G$ containing $e$. Is there a finite subset $B$ ...

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**1**answer

336 views

### representing an uncountable free group as a union of an increasing sequence of countable subgroups

Let $(G_\alpha)$ and $(K_\alpha)$ $(\alpha<\aleph_1)$ be strictly increasing chains of countable sets such that if $\alpha$ is a limit, then $G_\alpha=\bigcup_{\beta<\alpha}G_\beta$ and $K_\...

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336 views

### Example of a ring with non-finitely generated unit group?

The well known Dirichlet's unit theorem states that the unit group of a maximal order in a quadratic number field is finitely generated of rank blah blah blah. I think it's pretty naive to expect a ...

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83 views

### Character degrees of a finite group?

Let $G$ be a finite group, $R(G)$ be the solvable radical of $G$, and $G/R(G)$ be an almost simple group with socle $S\cong PSL_2(3^p)$, where $p$ is an odd prime. Also let $[G/R(G):S]=p$ and $\theta$ ...

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77 views

### Ordered group acting freely on partially ordered set

Let $(G, <)$ be a totally ordered group, and let $<$ be left-invariant. Let $G$ act (freely?) on a partially ordered set $(S, <)$, such that this group action preserves the ordering:
$$ s_1 &...

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184 views

### What is the complexity of determining if a knot group is $\mathbb{Z}$?

It is known from the work of Waldhausen that the isomorphism problem for knot groups is decidable. What is then:
The complexity of determining if a knot group is $\mathbb{Z}$? .i.e. same as the ...

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**1**answer

240 views

### A question about spectral properties of a non-amenable group

Let $G$ be a group generated by $a,b$ (for the sake of simplicity). Consider the element
$$S=a+b+a^{-1}+b^{-1}\in{\mathbb C}[G],$$
which may also be interpreted as an operator in $l^2(G)$ (by left ...

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211 views

### Amalgamated product of automatic groups

In Gersten's "Problems on Automatic Groups", Problem 14, he asks the following question: Let $G=A\ast_{C}B$ where $A$ and $B$ are automatic and $C$ is infinite cyclic. Is $G$ automatic?
Is this ...

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**1**answer

179 views

### On the general linear group of a vector space of infinite dimension

Let $F/\mathbb Q$ be a finite normal extension of the rational numbers. Let $V$ be an $F$-vector space of countably infinite dimension, and set $L=GL_F(V)$. Put moreover $L^*$ be the set of all ...

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106 views

### The finite extensions of $SL_2(q)$ [closed]

Let $G$ be a finite group such that $Z(G)\leq SL_2(q) \leq G $ and $G/Z(G) \cong PGL_2(q)$· Is there any information about the structure of $G$?

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120 views

### Tarski number is not a quasi isometric invariant, an example?

I know that Tarski number is not a quasi isometric invariant, i.e.
Let $G,H$ be two groups such that $G\sim_{QI} H$, then it is not necessary to have $T(G)=T(H)$.
But can you bring an example for ...

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75 views

### Software/program for finding relations $w\in \overline{R}$ given group presentation $\langle X\mid R\rangle$

Given a presentation of a finitely presented group $\langle X\mid R\rangle$, is there a program that can list all relations of certain length i.e. words $w\in \overline{R}$ with length $n$ ( $aba^{-1}$...

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290 views

### How many non-isomorphic groups share the same character table?

I have been thinking for a while about how to enumerate all finite groups. The classical way e.g. here would be to go via latin squares and then try to calculate how many of those obey associativity. ...

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**1**answer

210 views

### Extension-field subgroups of $\operatorname{GL}(n, K)$

$\newcommand{\op}[1]{\operatorname{#1}}\def\GL{\op{GL}}$I am interested in the subgroups of $\GL(n,K)$ of the form $\GL(n/s, E)$ for $E$ some extension field of $K$ of degree $s$. These arise in the ...