Questions about the branch of abstract algebra that deals with groups.
0
votes
0answers
1 view
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, ...
1
vote
0answers
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 ...
-1
votes
0answers
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 ...
6
votes
0answers
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 ...
33
votes
7answers
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 ...
8
votes
1answer
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 ...
-2
votes
0answers
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 .
7
votes
1answer
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 $|\chi(g)|=n$.
...
2
votes
1answer
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.
27
votes
1answer
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 ...
5
votes
1answer
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}...
1
vote
1answer
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-...
3
votes
0answers
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).
...
5
votes
1answer
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 ...
4
votes
1answer
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 ...
1
vote
0answers
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 ...
3
votes
1answer
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 ...
1
vote
0answers
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 ...
12
votes
0answers
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 ...
6
votes
1answer
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 ...
8
votes
2answers
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? ...
2
votes
1answer
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 ...
1
vote
0answers
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 $...
4
votes
1answer
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 ...
12
votes
1answer
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 ...
11
votes
0answers
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 $\...
3
votes
1answer
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 $...
7
votes
0answers
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, ...
4
votes
0answers
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 ...
3
votes
0answers
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 ...
2
votes
0answers
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{...
18
votes
2answers
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 ...
1
vote
1answer
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 $\text{Sub}(G) \cong {\cal P}(\kappa)$ ...
1
vote
1answer
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 ...
7
votes
0answers
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 ...
3
votes
2answers
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 ...
2
votes
1answer
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 $B$ ...
5
votes
1answer
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_\...
11
votes
2answers
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 ...
1
vote
0answers
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 $\theta$ ...
0
votes
1answer
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:
$$ s_1 &...
6
votes
1answer
184 views
What is the complexity of determining if a knot group is $\mathbb{Z}$?
It is known from the work of Waldhausen that the isomorphism problem for knot groups is decidable. What is then:
The complexity of determining if a knot group is $\mathbb{Z}$? .i.e. same as the ...
5
votes
1answer
240 views
A question about spectral properties of a non-amenable group
Let $G$ be a group generated by $a,b$ (for the sake of simplicity). Consider the element
$$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 ...
10
votes
0answers
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 ...
2
votes
1answer
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 ...
1
vote
0answers
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 $G$?
6
votes
0answers
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 ...
0
votes
0answers
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 $n$ ( $aba^{-1}$...
3
votes
0answers
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. ...
6
votes
1answer
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 ...
