Questions about the branch of abstract algebra that deals with groups.
4
votes
1answer
57 views
Is the Thompson group F locally indicable?
A group $G$ is called locally indicable if for any finitely generated subgroup $H \subset G$, there is a non-trivial homomorphism from $H$ to the real additive group $(\mathbb{R},+)$.
Is the Thompson ...
1
vote
1answer
123 views
the number of Sylow subgroups
Let $p>3$ be a prime number and $G$ be a finite group of order $2p(p^2+1)$.
Is it true that always the Sylow $p$- subgroup of $G$ is a normal subgroup of $G$?
As I checked by Gap it seems true.
...
4
votes
1answer
112 views
Rings with group of units cyclic of prime order
For what prime numbers $p$ there exists a ring with identity and exactly $p$ invertible elements ?
REMARK It can be shown that for $p=5$ there is no such ring, so I am wondering for what values of ...
3
votes
0answers
62 views
Is this still an open question? (Orthogonal Orthomorphisms/Complete Maps)
The motivation for my question came after reading this article:
http://cms.math.ca/cjm/v13/cjm1961v13.0356-0372.pdf
It is quite old so I hope someone here knows what advances have been made in this ...
5
votes
0answers
166 views
Tarski Monster group with prime 5
Does the Tarski Monster group with prime 5 exist? I know that for 2 and 3, the group does not exist, but what about 5?
3
votes
2answers
102 views
Maximal $k$-transitivity for a proper subgroup of $S_n$
Let $S_n$ be the permutation group on $n$ elements.
Denote by $K(n)$ the largest $k$ s.t. $S_n$ has a $k$-transitive subgroup (w.r.t. its action on the $n$-element set on which $S_n$ acts) different ...
2
votes
1answer
108 views
Fixed points of IA automorphisms
Let $F_n$ denote the free group on $n$ generators $x_1,\ldots , x_n$.
Recall that an element $\varphi\in\mathrm{Aut}(F_n)$ is an IA automorphism if it induces the identity on the abelianization ...
4
votes
1answer
116 views
Easy argument for “connected simple real rank zero Lie groups are compact”?
Let $G$ be a connected simple Lie group. It is known that if $G$ has real rank zero, then $G$ is compact.
Background: every connected (semi)simple Lie group $G$ (with Lie algebra $\mathfrak{g}$) has ...
1
vote
1answer
62 views
The action of graph automorphism of finite symplectic group on maximal subgroups
Let $G=Sp(4,2^f)$ with $f>1$. Based on the facts when $f$ is small, I would feel the following:
$G$ has two conjugacy classes of subgroups isomorphic to $SO^+(4,2^f)$. One is in Aschbacher's class ...
13
votes
0answers
112 views
Fundamental groups of reduced subgroup lattices
Let $G$ be a group. Its subgroup lattice, denoted $\Sigma G$, consists of all subgroups of $G$ partially ordered by inclusion. The topology of this poset is quite trivial, since it always has a ...
3
votes
1answer
99 views
Reference for embeddings of reflection groups (related to folding ADE Coxeter graphs?
There are a couple of indirect methods, using Lie theory or Springer's general theory of regular elements in (real, complex) reflection groups, to construct natural embeddings among certain Weyl ...
13
votes
2answers
340 views
Spin group as an automorphism group
Consider the real algebraic group $SO(p,q)$, this is the automorphism group of the vector space $\mathbb{R}^n$ of dimension $n=p+q$ over $\mathbb{R}$, endowed with the diagonal quadratic form with ...
4
votes
0answers
137 views
centralizer of the order 2^k cyclic permutation matrix over F_2
Let $C$ be the $2^k\times 2^k$-permutation matrix over $\mathbb{F}_2$ of the $2^k$-cycle. We needed to know the structure of its centralizer in $\mathrm{GL}_{2^k}(\mathbb{F}_2)$, and we computed it - ...
3
votes
3answers
192 views
Generalized free product of semigroups with amalgamated subsemigroups
Hanna Neumann in
[American Journal of Mathematics, 1948,
http://www.jstor.org/discover/10.2307/2372201?uid=2&uid=4&sid=21102497379451 ]
introduced a notion of generalized free product of ...
4
votes
2answers
175 views
Restricting the composition factors of subgroups of GL_m(Z/nZ)
For a positive integer $m$, let $\mathcal{A}(m)$ be the set of all integers $k \geq 5$ such that: there is a positive integer $n$ and a subgroup $G \subset \operatorname{GL}_m(\mathbb{Z}/n\mathbb{Z})$ ...
