# Tagged Questions

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

**3**

votes

**1**answer

24 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 on the unit circle.
Assume that the eigenvalues of $A$ are included in a circle arc of length ...

**1**

vote

**1**answer

60 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

**0**answers

44 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

**2**answers

250 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

**4**answers

248 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

**2**answers

149 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

**0**answers

63 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

**1**answer

123 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

**1**answer

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

**0**answers

61 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, ...

**14**

votes

**1**answer

708 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

**0**answers

54 views

### On the schur Multiplier of a group [on hold]

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

**2**answers

205 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

**0**answers

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

**1**answer

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

**2**answers

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

**2**answers

185 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 ...

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**0**answers

143 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

**1**answer

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**

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**0**answers

90 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

**1**answer

50 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**

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**0**answers

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

**2**answers

126 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**

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**4**answers

457 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

**0**answers

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**

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**0**answers

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 ...

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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 ...

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vote

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74 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

**0**answers

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

**1**answer

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

**2**answers

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

**2**answers

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

**0**answers

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 ...

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**0**answers

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

**1**answer

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 ...

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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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

333 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

**0**answers

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

**1**answer

185 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

**2**answers

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 ...

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vote

**0**answers

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

**1**answer

220 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

**0**answers

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

**0**answers

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

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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.

**-1**

votes

**0**answers

28 views

### Weight diagram for sl(3,C) [migrated]

I am extremely bad at asking questions.This is an attempt.
I am having a little bit of trouble understanding the procedure involved in constructing weight diagrams. Can someone sketch the basics of ...