# Tagged Questions

Questions on group theory which concern finite groups.

132 views

### Counting elements with certain word length in abelian groups

Given a (finite) abelian group $G = \langle S \mid R \rangle$, has the problem of counting the number of elements which can be expressed as a word (in $S$) of length $\leq k$ been studied? If so, ...
192 views

### Is there a way to find an efficient set of relations for presenting the subgroup generated by two matrices in $SL(2, q)$?

Given two elements $a, b \in SL(2, \mathbb{F}_q)$, is there a way to find an efficient presentation $$\langle x, y \mid \text{relations}\rangle$$ of the subgroup $\langle a, b \rangle$? My intention ...
134 views

### What are the 2-generated subgroups of the special linear group $SL(2, q)$ over a finite field?

What is the subgroup structure of the subgroups $\langle a, b\rangle$ where $a, b \in SL(2, q)$?
238 views

### Is every nonabelian finite simple group a quotient of a triangle group $(a,b,c)$ with $a,b,c$ coprime?

Here by triangle group $(a,b,c)$ I mean the group with presentation $$\langle x,y \;|\; x^a = y^b = (xy)^c = 1\rangle$$ In other words, for every finite simple nonabelian group $G$, do there exist ...
547 views

### Is there a structure theorem or group law for finite groups generated by two elements?

Say that $a, b \in G$ are two elements of a finite group $G$. Is there a structure theorem for the structure of $\langle a,b\rangle$? Is there a way to derive group laws for the group operation in the ...
174 views

### A representation of a finite group where every nonzero vector has a trivial stabilizer [duplicate]

What are the finite groups which admit a non-zero representation in char 0 where every non-zero vector has stabilizer equal to $\left<1\right>$? Cyclic groups of prime order is one obvious class....
219 views

### A class 3 group of order 243

Let G be a group of order $243=3^5$. We denote by $(G_i)$ its lower central series and assume that $G$ has class $3$ and that $|G:G_2|=|G_3|=9$. We assume moreover that the cubing map factors as a (...
4k views

### Why can't a nonabelian group be 75% abelian?

This question asks for intuition, not a proof. An earlier question, Measures of non-abelian-ness was thoroughly answered by Arturo Magidin. A paper by Gustafson1 proves that, for a nonabelian group, ...
207 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? ...
199 views

### Isotropic subspaces in a symplectic vectorspace over $GF(q)$

Let $V$ be a symplectic vectorspace of dimension $2n$, and $r\mid n$. Is this statement true?"There is an isotropic spread of $r$ dimensional subspaces in $V$". By an isotropis subspace I mean a ...
275 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 ...
150 views

140 views

114 views

255 views

### A simple group that its order divide order of an alternating group

Let $G$ be a simple group such that 1) $|G|\mid|\mathrm{Alt}_{p}|$ 2) $p\mid | G|$, and $p>13$ is prime. 3) $G$ hasn't any elements of order $rp$ for every prime number $r$. My question: (...
93 views

### Localized at $p$ integral representations of finite elementary $p$-groups

Let $C_p$ be a cyclic group of prime order $p$. Let $F=C_p^n=C_p\times\dots\times C_p$ ($n$ times). I would like to to classify finite dimensional representations of $F$ over ${\mathbb{Z}}$. However, ...
125 views

### about subgroup of general linear group [closed]

Thanks for any comments Let $G=GL_n(F)$ be general linear group over finite field $F$. Consider two isomorphic subgeoup $H_1,H_2$ of $G$ such that $H_i\cong GL_k(\bar{F})$, where $\bar{F}$ is an ...
115 views

### Finite groups normalizing a torus

Let $G$ be a semi-simple linear algebraic group over the complex numbers, e.g. the special linear group. Can you find an example of a finite sub-group $H$ of $G$ which does not normalize any maximal ...
109 views

114 views

### Does every connected vertex transitive graph on $n$ vertices (except for $C_n$) have minimum feedback vertex set of size $\Omega(n)$?

Feedback vertex set is a set of vertices whose removal leaves an acyclic graph. It is known that every vertex transitive graph on $n$ vertices has minimum vertex cover of size $\Omega(n)$. It is also ...
### Extensions of $SL(2,\mathbb{F}_q)$
Let $n = q(q^2-1)$ (the order of $SL(2,\mathbb{F}_q)$ I think). How many groups of order $2n$ contain $SL(2,\mathbb{F}_q)$ as a (necessarily normal) subgroup? Is this number known exactly? Seems ...