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

**12**

votes

**4**answers

644 views

### Determining if a matrix is orthogonal

Let g be an element of $GL_n(\mathbb C)$. We know that there are orthogonal groups $O(\beta)=\{X\in GL_n(\mathbb C) \mid X^t\beta X=\beta\}$ for any $\beta$, invertible symmetric matrix. Though these ...

**-2**

votes

**0**answers

28 views

### line complex in projective space [on hold]

In a paper of "R.H.Dye", which you can find here: http://link.springer.com/article/10.1007%2FBF02413785#page-1, I face with a mathematical object "line complex in projective space $PG(2n-1,q)$, I need ...

**18**

votes

**1**answer

366 views

### Group with finite outer automorphism group and large center

Does there exist a finitely generated group $G$ with outer automorphism group $\mathrm{Out}(G)$ finite, whose center contains infinitely many elements of order $p$ for some prime $p$?
A motivation is ...

**1**

vote

**1**answer

140 views

### Subgroup of Projective general linear group on complete discrete valuation ring

Let $R$ be a complete dvr and $k$ its residue field of positive characteristic.
Let $H$ be a finite subgroup of $PGL_2(k)$ such that the order of $H$ is prime with $char(k)$.
Is there some ...

**-3**

votes

**0**answers

39 views

### How can we find the presentation of a group? [on hold]

Is it possible that to find the presentation of a group $G$ such that it is extension of $\mathbb{Z}_2\times \mathbb{Z}_2$ by $\mathbb{Z}_2$?

**3**

votes

**1**answer

632 views

### Special automorphisms of extraspecial groups

Let $G$ be an extraspecial group of order $p^{2r+1}$ (where $p$ is an odd prime), and let $V$ be a faithful representation of $G$. Consider the normal subgroup $H$ of $Aut(G)$ consisting of all ...

**0**

votes

**1**answer

167 views

### Find all possible rational pairs of a parametric sextic and all cases where it is reducible for either parameter

Find all possible rational solutions pairs $(z,a)$ of the equation
$a^6 + a^4 (-18368 + 9184 z - 2912 z^2) + a^2 (61702144 - 61702144 z + 36814848 z^2 - 10694656 z^3 + 1748992 z^4) - z^2 ...

**-4**

votes

**0**answers

44 views

### Number of homomorphisms between finitely generated abelian group and a finite cyclic group [closed]

This is the situation:
Suppose we have a finitely generated group $G= \mathbb{Z}^r \times E$ with $E$ it's tortion subgroup and $m=$ exponent of $E$ i.e. the least natural number such that $x^m=1$ ...

**4**

votes

**1**answer

97 views

### Restricted Burnside Problem: Lower bound nilpotency class

Let $p$ be a prime and let $F$ be a free group of rank $d\geq 1$.
Kostrikin [1] proved that the $d$-generated Burnside group $B=B(d,p)=F/F^p$
of exponent $p$ has a maximal finite quotient
...

**1**

vote

**0**answers

76 views

### Bound on $g(n+1)/g(n)$ for Landau's function

I have read that for the Landau function $g(n)$ (http://mathworld.wolfram.com/LandausFunction.html), one knows that
$\lim_{n \rightarrow \infty} g(n+1)/g(n) = 1$
(stated, but not proved in "On ...

**0**

votes

**0**answers

206 views

### Whitehead group [closed]

Whitehead group (WG) is known for some groups (e.g. free abelian group, cyclic group, Braid group etc).
For example:
The Whitehead group of the trivial group is trivial.
The Whitehead group of a ...

**5**

votes

**1**answer

154 views

### Hyperbolic knot complement groups and relative dimension

Let $G$ be a the fundamental group of a hyperbolic knot complement.
Then $G$ is hyperbolic relative to a subgroup $P\cong \mathbb Z \oplus \mathbb Z$.
The knot complement has a $2$-dimensional spine ...

**4**

votes

**2**answers

241 views

### What is the motivation and purpose of the Floretion group?

When searching through the Oeis, I came across something called a floretion. Based on the context, it seems to be some sort of algebraic structure. I googled it and found nothing that explained their ...

**5**

votes

**2**answers

155 views

### PSL(2,p) as quotient of triangle groups

As a by-product of some Magma computations, I've observed that, for each prime $p$
such that ${\rm PSL}(2,\mathbb{F}_p)$ can be a quotient of the $(3,3,4)$-triangle group
(i.e. $p \equiv \pm 1 ...

**0**

votes

**1**answer

125 views

### Chinese remainder theorem for cyclic subfactor planar algebras

This post was inspired by an exchange with the indian woman mathematician Ajit Iqbal Singh.
The chinese remainder theorem can be stated as follows:
Let $n_1, \dots, n_r \ge 2$ be positive integers ...

**-4**

votes

**0**answers

54 views

### Cayley graph of dihedral group is isomorphic to which kind of graphs? [on hold]

Let D_{2n}= be dihedral group of order 2n. Also let D_{2n}= in which 1\ notin S=S^{-1}.
In this case Cay(D_{2n}, S) is isomorphic to which kind of graphs? This is my conjecture that this graph is ...

**2**

votes

**1**answer

112 views

### Asymptotic of min(#generators times diameter), for a Cayley graph of Sn

This post is a sequel of Diameter of symmetric group.
Let $\Sigma$ a generating subset of $S_n$, $\Gamma(S_n, \Sigma)$ the Cayley graph and $d_{\Sigma}$ the diameter of $\Gamma(S_n, \Sigma)$.
Let ...

**-2**

votes

**0**answers

61 views

### On cyclic decomposition of element in $S_n$? [closed]

Let $S_n$ be symmetric group and $x\in S_n$ be a permutation of $n$ numbers. Let $|x|=p$, where $n/2<p<n$ is prime. Consider $1^{t_1}2^{t_2}\ldots l^{t_l}$ to be the cyclic decomposition of $x$. ...

**6**

votes

**0**answers

426 views

### Inverse Galois Problem…and parallelizable vector fields?

Usual approaches to the inverse Galois problem start with realizations of a group $G$ over a larger field, and then try to specialize to ${\Bbb Q}$.
One could also start by building suitable objects ...

**3**

votes

**0**answers

81 views

### Application of finding shortest paths on Cayley graphs

For a fixed integer number $m$, Consider Cayley graph defined by all m-cycles in Symmetric group $Sym(n)$.
I know that for $m=2$,
there are some applications of finding shortest paths (or distance ...

**0**

votes

**0**answers

71 views

### Existence of non-trivial characters on compact abelian group [closed]

Does for every compact (compact metric) abelian group $(G, \odot )$ there exist a non-trivial homomorphism $\varphi : (G, \odot ) \to (\mathbb{C} , \cdot ) $ such that $|\varphi (g) |=1$ for all ...

**2**

votes

**1**answer

185 views

### Classes of finite resoluble groups which are (faithfully) representable by triangular matrices?

Let $G$ be a group, $k$ a field and $T(n,k)\subset Gl(n,k)$,
the group of invertible upper triangular $n\times n$ matrices.
I know that if $\rho : G\rightarrow T(n,k)$ is faithful
(i.e. into) then ...

**6**

votes

**3**answers

565 views

### Classification of groups in which the centralizer of every non-identity element is cyclic

In which classes of groups is it feasible to classify those groups in which the centralizer
of every non-identity element is cyclic?

**2**

votes

**2**answers

425 views

### What is the $\mathbb Z/2$-cohomology of $\mathrm B^n(\mathbb Z/2)$?

I would like to know the cohomology groups $\mathrm H^\bullet(\mathrm B^n(\mathbb Z/2);\mathbb Z/2)$. I assume that this is a standard computation, but I'm not sure where to look up the answer (and, ...

**0**

votes

**0**answers

72 views

### What is the diameter of the Cayley graph of $S_n$ w.r.t the generator $\{(12),(23), . . . ,(n − 1 n) \}$? [duplicate]

Given a symmetric group $S_n $ and the generator set : $S = \{(12),(23), . . . ,(n − 1 n) \}$ is there any closed form expression for the diameter of the Cayley graph generated by this set of ...

**3**

votes

**0**answers

237 views

### On Thompson conjecture

Let $N(G)$ be the set of conjugacy class sizes of finite group $G$. Let $G$ be a finite group for which $N(G)=N(A_n)$, where $A_n$ is the alternating group of degree $n$.
Let $p$ and $q$ be distinct ...

**1**

vote

**1**answer

174 views

### Abelianization of limit groups

Let $G_1$ and $G_2$ be limit groups, and let $C_1$ and $C_2$ be cyclic subgroups of $G_1$ and $G_2$, respectively.
Question:
If $G$ is the amalgamated product of $G_1$ and $G_2$ with amalgamated ...

**22**

votes

**0**answers

387 views

### Next steps on formal proof of classification of finite simple groups

While people are steaming ahead on finessing the proof of the classification of finite simple groups (CFSG), we have a formal proof in Coq of one of the first major components: the Feit-Thompson ...

**32**

votes

**1**answer

687 views

### Transitivity on $\mathbb{N}_0$ — a 42 problem

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$.
Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class
transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the ...

**1**

vote

**0**answers

52 views

### A discrete presentation for a free prop-$p$-group

Let $n\geq 1$ be an integer and $p$ a prime. Suppose that $\mathcal{F}(n,p)$ is the free prop-p-group of rank $n$.
Question: For each pair $(n,p)$, is it known a discrete free group $\mathfrak{F}$ ...

**0**

votes

**2**answers

217 views

### Representation Theory of $U(N)$

(1) Is it true that the category of representations of $U(n)$ is equivalent to the category of representations of $SU(N) \times U(1)$? If so, how is it proved, or what is a good reference. (I guess ...

**0**

votes

**0**answers

121 views

### Representation of finite group

Let $\Gamma$ be a finite index subgroup of $SL_2(\mathbb{Z})$. Let $\pi$ be a natural covering from $\Gamma\backslash\mathbb{H}$ to $SL_2(\mathbb{Z})\backslash\mathbb{H}$. Denote by $\text{Deck}(\pi)$ ...

**1**

vote

**1**answer

97 views

### Anyone have missing reference list - Kerber “Representations . . . I”

My copy of Kerber's Representations of Permutation Groups I is missing the pages containing the references. Anybody got a copy that shows such?

**32**

votes

**1**answer

895 views

### Does a finite simple group of order divisible by $60$ have $A_{5}$ as a subgroup?

The title says it all really. I know that in some "small" simple groups such as ${\rm PSL}(2,p)$ it is known that if the order is divisible by $60,$ then $A_{5}$ does occur as a subgroup ( this is ...

**3**

votes

**0**answers

116 views

### pro-p dense subgroup in the free group

Let $G_p$ be pseudovariety of all finite $p$-group. The pro-$p$ topology on a group $G$ is the unique group topology such that the set of normal subgroups $N$ with $G/N$ in $G_p$ is a fundamental ...

**0**

votes

**0**answers

153 views

### Can ugly groups have derived length 3?

Definitions: All groups referred to are finite solvable. Call such a group good if it can be constructed from the trivial group using central extensions and split extensions, call a group bad if it ...

**13**

votes

**1**answer

662 views

### Finitely generated group with $\aleph_0<X_G<2^{\aleph_0}$ normal subgroups?

Let $X_G$ be the number of normal subgroups of a group $G$. Are there examples of finitely generated groups $G$ where it is consistent to have $\aleph_0<X_G<2^{\aleph_0}$ normal subgroups? Also ...

**3**

votes

**1**answer

138 views

### A possible presentation with 2 generators and 2 relators for $C_4 \cdot D_8$

Is there a presentation with two generators and two relators for the group $C_4 \cdot D_8$?
This group is of order 32 and its IdSmallGroup in GAP is [32,15].
Also it has the following presentation ...

**3**

votes

**1**answer

130 views

### minimal polynomial of unipotents in orthogonal group

Consider split orthogonal group O(2l) over a field of characteristic zero. We may assume the matrix of bilinear form to be $\begin{pmatrix} O&I\\ I&0\end{pmatrix}$. Let u be a unipotent in ...

**2**

votes

**2**answers

129 views

### Finite groups which have elements $g$ of order $pq$ such that the sizes of the conjugacy classes of $g^p$, of $g^q$ and of $g^{q-p}$ coincide [closed]

Let $G$ be a finite group and let $g \in G$ be an element of order $pq$, where $p < q$ are
prime numbers. Denote by $g^G$ the conjugacy class of $g$ in $G$. Under which conditions
does the ...

**4**

votes

**2**answers

628 views

**7**

votes

**1**answer

222 views

### Countable group with uncountable number of subgroups $< 2^{\aleph_0}$

Is it consistent that there is a countable group $G$ such that the cardinality of the set of subgroups of $G$ is uncountable, but strictly less than $2^{\aleph_0}$?

**7**

votes

**2**answers

498 views

### Avoiding countable subgroups of general uncountable groups

The following problem is a general form of another problem (motivation is available there). Initially, the problems were posted together, but the first one is solved below, a solution that does not ...

**3**

votes

**1**answer

250 views

### simultaneous action of GL(n) on the matrices

Consider the action of GL(n,k) on the set MxM where M is the set of all n-by-n matrices over k given by $g.(h,l) \mapsto (ghg^{-1}, glg^t)$. Individually these actions are well studied and good ...

**2**

votes

**0**answers

112 views

### The defining relations for a subgroup of $SL(2,Z)$

$SL(2,Z)$ generated by $T=\begin{pmatrix} 1 & 1\\ 0& 1\\ \end{pmatrix}$ and $S=\begin{pmatrix} 0 & 1\\ -1& 0\\ \end{pmatrix}$ has the following defining relations
$S^2=(S T)^3=C,\ ...

**2**

votes

**0**answers

86 views

### Minimal number of defining relators of a finite $p$-group on a minimal generating set

What is the state-of-art of the following question?
Let $p$ be any prime number. For any finite $p$-group $G$, let $r_G$ denote the minimum number of defining relators in all presentations of $G$ ...

**50**

votes

**16**answers

5k views

### Solving algebraic problems with topology

Often, topologists reduce a problem which is - in some sense - of geometric nature, into an algebraic question that is then (partiallly) solved to give back some understanding of the original problem.
...

**1**

vote

**0**answers

163 views

### Are the finite groups inclusions, almost all relatively cyclic?

Definition: An inclusion of finite groups $(A \subset B)$ is relatively cyclic if $\exists b \in B$ such that $\langle A,b \rangle = B$.
Definition: Two inclusions of finite groups are equivalent, ...

**2**

votes

**0**answers

84 views

### pro-p topology on a free group

James Howie in the paper "The p-adic topology on a free group:a counterexample" showed that in the free group $F$ generated by $x$ and $y$,if $a=xy^2$, $b_1=x^{-2}y^{-3}$ and $b_2=x^{-2}(xy)^5$, then ...

**17**

votes

**2**answers

517 views

### Is there a big solvable subgroup in every finite group?

Definition: Let $G$ be a group, and let $H \leq G$ be a subgroup. We say that $H$ is big in $G$ if for every intermediate subgroup $H \leq L \leq G$ there exists some $x \in L$ such that $\langle H ...