Tagged Questions
Questions about the branch of abstract algebra that deals with groups.
-4
votes
0answers
44 views
Why is it true that write $\bar{G}=\bar{H}\bar{K}$ [on hold]
Let $G=HK$ and $N$ be a normal subgroup of $G$. Also let $\bar{G}=G/N$. Why is it true that write $\bar{G}=\bar{H}\bar{K}$
19
votes
1answer
380 views
Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$
Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle.
Assume that the eigenvalues of $A$ are included in a circle arc of ...
2
votes
2answers
117 views
Bound for the Frattini subgroup of a $p$-group
Assume that $G$ is a finite $p$-group, $p$ odd, with a non-trivial elementary abelian Frattini subgroup. Then both $\Phi(G)$ and $G/ \Phi(G)$ are vector spaces over $\mathbb{F}_p$. Is it possible to ...
0
votes
0answers
49 views
Characteristic subgroups of the limit group
Let $\{ G_i \}_{i=1}^\infty$ be a direct spectrum of groups with respect to embeddings $\varphi_i:G_i \mapsto G_{i+1}$, $i \in \mathbb{N}$, and let $G$ be the limit group of this spectrum. Suppose ...
11
votes
2answers
272 views
Subgroups of $SL_3(\mathbb{Z})$ that are finitely generated, Zariski-dense, infinite index, and torsion-free
My question stems from Misha's answer of a MathOverflow question. Misha supplied the following question in his answer:
Open question: Does there exist a finitely generated Zariski-dense ...
2
votes
4answers
250 views
Variations to Cayley's Embedding Theorem for Groups
Early in a course in Algebra the result that every group can be embedded as a subgroup
of a symmetric group is introduced. One can further work on it to embed it as a subgroup of a suitable (higher ...
4
votes
2answers
155 views
Can finitely generated subgroups of limit groups be detected in free group quotients?
In Henry Wilton's excellent paper "Hall's Theorem for limit groups" (Geom. Funct. Anal. 18, pp. 271–303, 2008 ) he proves the following result (see his paper or here and here for the relevant ...
1
vote
0answers
68 views
$p$-groups with $\Omega_1(G)\leq\Phi(G)$ [on hold]
Let $G$ be a finite $p$-group with $\Omega_1(G)\leq\Phi(G)$.
What do we have information about this group?
1
vote
1answer
125 views
What are finite groups $H$ such that $H^n(H,\mathbb{Q/Z}) \cong H_n(G,\mathbb{Z})$?
Let $G$ be a finite group and $G^{\prime}$ be its commutator subgroup. Let $\mathbb{Z}$ and $\mathbb{Q}$ denote the integers and rationals. $\mathbb{Z}$ and $\mathbb{Q/Z}$ treated as trivial ...
4
votes
1answer
189 views
Normal Subgroup Growth
Let $F$ be a free group on $d$ generators.
Denote by $F_{k}$ the $k$-th term in $F$'s derived series. Put $G = F/F_k$. What is the normal subgroup growth of $G$?
Explicitly, for each natural number ...
3
votes
0answers
62 views
Cancellations in products of two elements of a hyperbolic group
Let $G$ be a non-abelian free group with the standard generating set and the corresponding word metric. If we take two elements $g,h\in G$ and compute their product $gh$, some letters might cancel, ...
15
votes
1answer
786 views
How to prove that every polynomial in an infinite family is irreducible over Q?
Consider the bivariate polynomial $$p(X,Y) = X^5 - (2 Y + 1) X^3 - (Y^2 + 2) X^2 + Y (Y-1) X + Y^3.$$ For every integer $y \ge 4$, I conjecture that $p(X,y)$ is irreducible in $\mathbb{Q}[X]$. How can ...
0
votes
0answers
54 views
On the schur Multiplier of a group [closed]
Let $G$ be a finite simple group and its Schur multiplier is 2.
Is it true that if ${M\over K}\cong G$ and $|K|=2$, then $M\cong {\Bbb Z}_2\times G$ or $M\cong 2.G$, according to the symbols in the ...
10
votes
2answers
209 views
Algorithms in hyperbolic groups
I'm stuck in some algorithms in hyperbolic groups, which may be rather simple.
Let $G$ be a hyperbolic group given by a finite presentation. It is known that the hyperbolicity constant $\delta$ can ...
2
votes
0answers
90 views
Inn characteristic in Aut [migrated]
If $G$ is a centerless group then is $\mathrm{Inn}(G)$ necessarily characteristic in $\mathrm{Aut}(G)$?
The condition of being centerless is necessary as $D_8$ provides a counterexample otherwise.
3
votes
1answer
61 views
H S class operator and its equality
$A \in S(K)$ iff $A$ is a subalgebra of some member of $K$
$A \in H(K)$ iff $A$ is a homomorphic image of some member of $K$
It is trivial to see the containment $SH \leq HS$. Taking a simple ...
4
votes
2answers
227 views
Simple groups and words
Let S be a finite simple nonabelian group, w a word in a finite number of variables which is not a power of another word. Must there be a substitution of elements of S in w such that the resulting ...
3
votes
2answers
187 views
Schreier's index formula
A finitely generated group G is said to satisfy Schreier's index formula if for every subgroup H of index k in G we have: d(H) - 1 = k(d(G) - 1). For example, a finitely generated free group satisfies ...
9
votes
0answers
146 views
+100
$2$-group with two isomorphic normal subgroups of index $4$ with non-isomorphic quotients
Hypothesis: Let $P$ be a finite $2$-group with two isomorphic normal subgroups $M$ and $N$ such that $P/M\cong C_4$ (the cyclic group of order $4$) while $P/N\cong C_2^2$. By the lattice theorem, ...
1
vote
1answer
73 views
Relations between Arboreal Group Theory and Tree Group Actions?
By a tree group action, we mean an action of a group $G$ over the infinite regular binary tree $T_2$ such that for each $g \in G$, the mapping $x \rightarrow g.x$ is an automorphism of $T_2$; these ...
4
votes
0answers
91 views
The Alexander-Conway polynomial: from knots to braids?
The Alexander-Conway polynomial was the first knot invariant to be discovered, as far back as 1923 according to this link. Given that knots can be expressed in terms of quasi-toric braid closures, it ...
0
votes
1answer
51 views
Isometry group of an integer as of the corresponding $\Omega(n)$-parallelotope
Let $\prod_{i}p_{i}^{a_{i}}$ be the prime factorization of a positive integer $n$ and let's consider the $\Omega(n)$-parallelotope built with, for all $i\in I$, $a_{i}$ pairwise orthogonal vectors of ...
9
votes
0answers
174 views
Non split extension isomorphic (as a group) to a split extension
$\def\Z{\mathbb{Z}}$
Let $A$ be a finite abelian group and $G$ a finite group acting on $A$.
Then the extension $0 \to A \to E \to G \to 1$ is splits if and only if the corresponding $2$-cocycle is ...
4
votes
2answers
129 views
Embedding a linearly ordered free monoid into a linearly ordered group
A linearly ordered (shortly, l.o.) monoid is a triple $\mathbb M = (M, \cdot, \le)$ for which $(M, \cdot)$ is a (multiplicatively written) monoid and $\le$ is a total order on $M$ such that $xy < ...
7
votes
4answers
462 views
Conjugation Quandles and… “Quandle-Groups”? From quandles to Groups
This question is already asked MathSE
A quandle $(Q,*,/ )$ is a idempotent right-distributive and right invertible structure.
1) $a*a=a$
2) $(a*b)*c=(a*c)*(b*c)$
3) $(a*b) ...
3
votes
0answers
110 views
automorphism of finitly generated group
Let G be a finitely generated group , then can we say that the group of automorphisms of G is also finitely generated .If yes what is the relation between the number of generators.If not under what ...
5
votes
0answers
190 views
Characters and conjugacy classes [migrated]
This comes up in reading David Speyer's answer to this question. Given a finite group $G$ and two non-conjugate elements $x, y,$ how does one construct a unitary representation $\rho$ of $G$ such that ...
0
votes
0answers
83 views
Free Groups and Automatic Structures [migrated]
I would like to know How can we prove that free groups are automatics? and is it possible to determine the number of states that would have a shortlext automatic structure (Word Acceptor and ...
1
vote
0answers
75 views
Two $p$-groups whose automorphism groups have isomorphic Sylow $p$-subgroups
Fix a prime $p$, and let $M$ be the unique nonabelian group of order $p^3$ and exponent $p$. Let us denote by $E_n$ the elementary abelian group of rank $n$.
Is it true that $\operatorname{Aut}(M ...
1
vote
0answers
41 views
On generating Euler Square of index q, q-1 (where q is any prime power)
Can somebody help me in generating Euler Square of index q, q-1 (where q is any prime power). Also, kidly tell me if there is any code availeble for generating the stated Euler Square. The details of ...
1
vote
1answer
65 views
Control of $p$-extensions by subgroups of index coprime to $p$
Let $G$ be a finite group and let $M$ be a $G$-module that is a finite abelian $p$-group. Suppose we have extensions
$1 \rightarrow M \rightarrow E_1 \rightarrow G \rightarrow 1$
and
$1 ...
1
vote
2answers
200 views
Lattices in general totally disconnected locally compact groups
Besides automorphism groups of trees and buildings, I was wondering if the lattices in general totally disconnected locally compact groups have been studied in the literature? I appreciate if you ...
4
votes
2answers
102 views
Reduction of different RG lattices to kG modules
Every book on modular representation theory of finite groups introduces p-modular systems and describes how to reduce an ordinary representation $U$ to obtain one in characteristic p (call it ...
-2
votes
0answers
93 views
Given a set of generators of a group G, is there a method to find a presentation for G using those generators? [migrated]
Suppose I have a group $G$, which I know is finitely presentable and infinite. (In particular, I have a presentation for it, though not the one I want).
Suppose I have a small list of generators ...
0
votes
0answers
97 views
Orbits of stabilizer of two points in a 2-transitive permutation group
I was doing something which needs to know sizes of all orbits of the stabilizer of two points in a 2-transitive permutation group. Since all 2-transitive permutation groups are known ans so are their ...
4
votes
1answer
196 views
Group extensions isomorphic as groups
Let $G$ be a group and $A$ a $G$-module. It well know that there is a group isomorphism between the second cohomologoy group $H^2(G,A)$ and the abelian group $OpExt(G,A)$ of classes of extension ...
0
votes
0answers
133 views
Induced graphs of cayley graph
I have a Cayley graph $Cay(G,S)$, its group presentation $G=< S | R >$ and it is a metric graph by assigning a length equal to 1 to each edge. I also have an induced subgraph of that Cayley ...
2
votes
0answers
93 views
An equivariant Hahn Embedding Theorem?
The Hahn Embedding Theorem asserts that for any (linearly) ordered abelian group $\Lambda$, there exists a linearly ordered indexing set $\Omega$ such that $\Lambda$ admits an order-preserving group ...
5
votes
1answer
204 views
In which fixed-point free representations is the sum of every 3 elements invertible?
A representation $\rho:G\to GL_k(\mathbb{F})$ is called fixed-point free if for every $1\neq g\in G$ and every $0\neq v\in \mathbb{F}^k$, $\rho(g)v\neq v$. Stated differently, it is a representation ...
6
votes
1answer
205 views
Primitive, non-2-transitive groups with very large orbitals?
Let $G$ be a transitive permutation group on a set $X$ with $n$ elements. Assume that $G$ is primitive, i.e., $G$ preserves no non-trivial partition of $X$. Assume as well that $G$ is not ...
1
vote
1answer
185 views
On the Complement of a subgroup
This question was asked in
http://math.stackexchange.com/questions/729648. Since I did not get any answer I am asking it here.
In an answer in Mathoverflow I see an answer but I could not ...
15
votes
1answer
334 views
Finite groups $G$ so that $G$ has exactly two subgroups of a given order
Is there a finite group $G$ and a divisor $d$ of $|G|$ so that $G$ contains exactly two subgroups of order $d$?
The motivation for this question is an old qual problem (see ...
3
votes
0answers
99 views
Criterion for a subgroup of $PSL_2(\mathbb R)$ to be Fuchsian
Let $\Gamma$ be a finitely generated subgroup of $PSL_2(\mathbb R)$.
I'm looking for effective criteria (idealy necessary and sufficient but just necessary would be a good start) ensuring that ...
6
votes
1answer
186 views
Topological groups defined by completely disconnected subgroups
Can you define a group topology on a group by specifying which subgroups should be discrete with respect to that topology (where a subgroup $S$ of $G$ is discrete if each $s\in S$ has an open ...
4
votes
2answers
166 views
Are the following Cayley digraphs Hamiltonian?
Consider the Cayley graphs $A'G_n$ on the alternating group $A_n$ with generating set $S = \{(1i2) : 3 \leq i \leq n\}$, for $n \geq 4$.
See the following page on Alternating Group Graphs for ...
1
vote
0answers
126 views
Existence of homogeneous single chain compositions of a given maximal subfactor?
All the subfactors here are irreducible inclusion of hyperfinite II$_1$ factors.
A subfactor $(N \subset M)$ is Homogeneous Single Chain ($HSC$) if its lattice of intermediate subfactors is a ...
9
votes
1answer
221 views
Multiply transitive groups, continued
This is related to this question. It is well-known that $S_n$ and $A_n$ are the only six transitive permutation groups, and it is likewise well-known that the proof of this requires the classification ...
3
votes
0answers
52 views
Double coset relation for unique intermediate subgroup (with homogeneity)
Let $G$ be a group and $H$ a subgroup such that there is a unique (non-trivial) intermediate subgroup $K$ (i.e. $H < S < G$ implies $S=K$) and $(H \subset K) \sim (K \subset G)$ (homogeneity) ...
1
vote
0answers
45 views
Minimal generating sets of free algebras of varieties
Let $V$ be a variety and $F$ be a relatively free algebra in $V$. Suppose $X$ is a minimal generating set for $F$. Under what conditions we can deduce that $X$ is a free basis of $F$?
1
vote
0answers
67 views
Groups of order 4p2 [migrated]
Dear Members of Mathoverflow, I am interested about a Fact (if it is right) of the structure of groups of order 4p^2. Let G be a nonabelian groups of order 4p^2, classify all this groups.
