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

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3
votes
1answer
338 views

Is a non-trivial finite perfect group of order 4n?

A finite group $G$ is perfect if $G = G^{(1)} := \langle [G,G] \rangle$, or equivalently, if any $1$-dimensional complex representation is trivial. Question: Is a non-trivial finite perfect group of ...
0
votes
0answers
68 views

What is the significance of the eigendecomposition of a Cayley table? [on hold]

Treating the Cayley table of a group $G$ as a matrix $M_g$, one notices interesting things about its eigendecomposition. For instance, for the symmetric groups $\{S_n\}$, the rank of the Cayley table ...
2
votes
0answers
167 views

Does knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ sometimes offer a significant advantage in finding $r$?

Is there a cyclic group $\mathcal G$ with generator $g$ for which the discrete log problem is assumed to be hard, but knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ for random $r$ makes finding $r$ easy? ...
-2
votes
0answers
33 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 ...
-3
votes
0answers
45 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 $\mathbb{Z}_2$?
13
votes
4answers
680 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 ...
1
vote
0answers
81 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 ...
4
votes
1answer
104 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 ...
4
votes
2answers
243 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
2answers
166 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 ...
5
votes
1answer
160 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 ...
1
vote
2answers
207 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 ...
-4
votes
0answers
56 views

Cayley graph of dihedral group is isomorphic to which kind of graphs? [closed]

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
0answers
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 $x$. ...
3
votes
0answers
83 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 ...
2
votes
1answer
113 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 ...
-4
votes
0answers
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 $x^m=1$ ...
0
votes
0answers
207 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 ...
0
votes
1answer
127 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 ...
0
votes
0answers
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
0answers
239 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 ...
22
votes
0answers
391 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 ...
1
vote
1answer
176 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 ...
2
votes
2answers
426 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, ...
1
vote
0answers
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
0answers
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 $\text{Deck}(\pi)$ ...
0
votes
2answers
220 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 ...
1
vote
1answer
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?
0
votes
0answers
154 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 ...
3
votes
0answers
121 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 ...
3
votes
1answer
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 ...
2
votes
1answer
186 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 ...
2
votes
2answers
130 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 ...
7
votes
1answer
224 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 $2^{\aleph_0}$?
2
votes
0answers
113 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
0answers
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 $G$ ...
3
votes
1answer
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 ...
3
votes
1answer
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 ...
32
votes
1answer
690 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 ...
2
votes
0answers
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 ...
1
vote
0answers
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, ...
1
vote
1answer
124 views

Groups with many vanishing elements

It is well-known that every non-linear character $\chi$ of a finite group $G$ vanishes on some elements of $G\setminus Z(\chi)$. The question is What can be said about a finite group $G$ for which ...
2
votes
1answer
159 views

Groups with $G^n \cong G$ for some integer $n$ [duplicate]

Which integers $n>2$ have the following property? There is a group $G$ such that $G^n \cong G$; and for all integers $k$ with $1<k<n$ we have $G^k\not \cong G$.
7
votes
2answers
499 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 ...
10
votes
1answer
339 views

Origin of group theory problem (bound on number of Sylow subgroups)

This problem (prove that the number of Sylow subgroups of a finite group $G$ is bounded by $\frac{2}{3}|G|$) posted on MSE proved rather difficult to solve. The OP has been silent about where the ...
1
vote
0answers
110 views

Finite quotients of an infinite product of finite groups

Let $G$ be a finite group. Consider the direct product $\Gamma=\prod_{i=1}^{\infty}G$ of (countably) infinitely many copies of $G$. For every finite set of numbers $\{i_1,\ldots,i_n\}$ we have the ...
1
vote
0answers
33 views

Different descriptions of the Baumslag-Solitar groups using affine groups [migrated]

On page 101 of this paper of Laurent Bartholdi which is a online documentation of the "fr" package of GAP which allows GAP to manipulate groups generated by automata ...
0
votes
0answers
63 views

Maximal subgroups which are not open in pro-2 groups

Is it true that a pro-2 group $G$ with $G/\Phi(G)\cong C_2\times C_2$ may have a maximal subgroup which is not open? Motivation: The Frattini subgroup of a profinite group by definition, is the ...
17
votes
2answers
518 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 ...
-2
votes
1answer
81 views

Can every non-discrete topological group G be algebraically generated by a nowhere dense subset?

Is there somone help me to show that if this problem have positive Answer : Problem :Can every non-discrete topological group G be algebraically gen- erated by a nowhere dense subset ? Thank ...