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

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votes

**1**answer

64 views

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

Let $G$ be a simple group such that
1) $|G|\mid|\mathrm{Allt}_{p}|$
2) $p\mid |
G|$, and $p>13$.
3) $G$ hasn't any elements of order $rp$ for every prime number $r$.
My question: (without ...

**1**

vote

**0**answers

36 views

### A subgroup of corank 1 in a free group contains a primitive element?

Let $F$ be the free group on $\{x_i\}_{i=1}^\infty$, and let $H \leq F$ be a subgroup with $\langle H \cup \{x_1\} \rangle = F$. Must there be a free basis $B$ of $F$ for which $B \cap H \neq ...

**3**

votes

**0**answers

147 views

### Primitive elements in a free group

Let $F$ be a free group of rank $\aleph_0$ and let $H$ be a subgroup for which there exists some $a \in F$ such that $\langle H \cup \{a\} \rangle = F$. Must there be some free basis $B$ for $F$ and a ...

**0**

votes

**0**answers

56 views

### growth series of groups [on hold]

As I know, in the literature there are formulas for groth series of direct product, free product and free product with amalgamation and graph product of groups.
Is there any formula that gives groth ...

**9**

votes

**2**answers

432 views

### Can any finite lattice be realized as an intermediate subgroups lattice?

Let $G$ be a finite group and $H$ a subgroup.
Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$.
Question: Can any finite lattice be realized as ...

**3**

votes

**1**answer

84 views

### nilpotent of class 2 free product

How is the nilpotent of class 2 (nil-2) free product of groups defined?
I came across this construction reading the following paper.
Alan H. Mekler (1981). Stability of nilpotent groups of class 2 ...

**3**

votes

**1**answer

128 views

### Preprint by Wall on Sjogren's theorem

In their account http://dx.doi.org/10.1016/0022-4049(87)90048-X of Sjogren's theorem, Cliff and Hartley refer to two articles:
[9] B. Hartley, A note on a lemma of Sjogren relating to. dimension ...

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votes

**1**answer

129 views

### Must a group of defficiency > 1 be nonabelian?

Let $F$ be a free group of finite rank $n > 2$, and let $S \subseteq F$ be a subset of cardinality at most $n-2$. Denote by $S^F$ the normal subgroup of $F$ generated by $S$. Must $F/S^F$ be ...

**3**

votes

**3**answers

177 views

### Enumerating cosets of the modular group

Suppose we chose the generators $f(z) = z+1, g(z) = z-1, h(z) = -1/z$ for the modular group $\Gamma$ (i.e. the group of fractional linear tranformations $z \mapsto (az +b)/(cz+d)$ with $a,b,c,d \in ...

**5**

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**0**answers

86 views

### Example of a torsionfree group satisfying a cohomological condition

Let us call a finitely generated group $G$ cohomologically rich if for each $k \geq 0$, we can find a subgroup $G'$ and a prime $p$ such that $H^k(G';\mathbb F_p) \neq 0$. Examples which come to mind ...

**0**

votes

**1**answer

133 views

### Consequences of Serre's property FA

Proposition 21 of Serre's Trees:
Let G be a group with property FA. If G is contained in an amalgam then G is contained in a conjugate of one of the amalgam's factors.
Can anybody help with this ...

**3**

votes

**0**answers

62 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, ...

**9**

votes

**2**answers

507 views

### Regular elements in the torus of a group of Lie type

Let $G$ be a simple linear algebraic group, and $F$ a Frobenius map, i.e. some power of $F$ is the standard Frobenius map which raises matrix entries to the $q$-th power. Then $G^F$ is a group of Lie ...

**3**

votes

**2**answers

291 views

### Classes of finitely generated groups for which it is known whether they contain periodic groups

Question: For which "interesting" classes of finitely generated groups is
it known whether every infinite group in the class has an element of infinite order?
Some examples:
For finitely ...

**4**

votes

**1**answer

159 views

### Finite p-groups and their fibered products

Is every finite $p$-group an epimorphic image of a fibered product of two finite $p$-groups which can be generated by $2$ elements?

**4**

votes

**1**answer

108 views

### K-fellow traveler property and automatic structure

I have been reading several articles about automatic groups and metric spaces of negative curvature. However it is not clear for me the relationship between automatic groups, hyperbolcity and the ...

**1**

vote

**1**answer

80 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 ...

**46**

votes

**5**answers

2k views

### Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?

Choose unit quaternions $q_0, q_1, q_2, q_3, q_4$ that form the vertices of a regular 4-simplex in the quaternions. Assume $q_0 = 1$. Let the other four generate a group via quaternion ...

**9**

votes

**0**answers

215 views

### Proof of Cauchy's Theorem from Group Theory - Generalizable?

There are many proofs for Cauchy's Theorem from group theory, which states that if a prime $p$ divides the order of a finite group $G$, then $\exists g\in G$ of order $p$.
Recently I've encountered ...

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votes

**0**answers

87 views

### Generating free groups by small subgroups and an element

Let $F$ be a free group of countably infinite rank, and let $L \leq F$ be a finite index subgroup. For some prime $p$ and $k \in \mathbb{N}$, let $\sigma \colon L \to ...

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votes

**0**answers

49 views

### relation between characters [on hold]

My assumption: $ H $ is a subgroup with index $ m $ in the finite group $ G $ & $ F $ is an algebraic closed field of characteristic zero & $ \chi $ is an irreducible $ F $-character of $ G $ ...

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votes

**0**answers

31 views

### $ \mathbb{C} $-character table of $ D_{14} $ [closed]

Is there any reference where I can find the $ \mathbb{C} $-character table of the dihedral group $ D_{14} $?

**5**

votes

**1**answer

283 views

### Elementary equivalence of the direct product and direct sum of groups

It is well-known that the direct product of any family of abelian groups
is an elementary extension of the direct sum of the family
(see e.g. Lemma A.1.6 in the book `Model Theory' by W. Hodges,
...

**10**

votes

**2**answers

443 views

### History of Tarski's problems on free groups

As is known, Tarski posed his questions about first-order theories of non-abelian free groups around 1945. However, the questions were not published in his papers or books.
What is the original ...

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votes

**0**answers

65 views

### About Frobenius groups [closed]

Let us consider these conditions:
$ G $ is a Frobenius group, $ N= $ kernel of $ G$ and $ L\lhd G $.
Now I want to know why either $ L\leqslant N $ or $N\leqslant L $?
thanks

**5**

votes

**1**answer

80 views

### Sum identities with immanants

For $\chi$ being an irreducible character of the symmetric group $S_n$ and being $M$ a complex $n\times n$-matrix, I would like to show
$$
\sum_{\sigma, \rho \in S_n} \overline{\chi(\sigma)} ...

**1**

vote

**1**answer

73 views

### Are Carter subgroups nilpotent projectors?

R. Carter prooved that in finite soluble groups $G$ Carter subgroups $C$ exist and that they are conjugated. Furthermore they are exactly the nilpotent projectors: For every normal subgroup $N$ of ...

**4**

votes

**2**answers

266 views

### Relation between Associative algebra and group algebra

Let $A$ be an associative algebra over a filed $k$.
Q) What are the condition we can impose on $A$ such that there exists a $G$ such that $A=k[G]$, the group algebra generated by $G$?
I am ...

**2**

votes

**2**answers

237 views

### Magnus' embedding theorem

I am looking for a (preferably modern) reference to the following old result of Magnus.
Let $F$ be a free group of finite rank and
$$ F_1 = [F,F], F_2 = [F_1,F_1], \dots , F_{n+1} = [F_n,F_n], \dots ...

**2**

votes

**0**answers

64 views

### Splitting over infinite generated abelian subgroup?

Recall that a group is CSA if all its maximal abelian subgroups are malnormal.
Question 1: Can a finitely generated CSA group splits (as graph of groups) over an infinitely generated abelian ...

**11**

votes

**0**answers

162 views

### Group with finite out 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

87 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 ...

**6**

votes

**1**answer

105 views

### Continuity of conjugation actions of Polish groups

Let $G$ and $H$ be Polish groups and let $\psi: G \rightarrow H$ be a continuous injective homomorphism such that $\psi(G)$ is normal in $H$. Then $H$ acts on $G$ by conjugation via $\psi$, in other ...

**5**

votes

**1**answer

295 views

### Finding a basis for the (linear combinations) span of a matrix group, efficiently?

I have an algorithm whose bottleneck is the following task:
Let $\mathbb{F}$ be a finite field.
Given a set of $k$ invertible matrices $g_1,\dots,g_k\in GL_n(\mathbb{F})$, let
$G=\langle ...

**7**

votes

**2**answers

644 views

### Translation length functions of non-simplicial trees

Let $G$ be a finitely generated group. By a theorem of Culler and Morgan, the set of non-abelian (not necessarily simplicial) minimal $\mathbb{R}$-trees with isometric $G$-action injects into the ...

**1**

vote

**1**answer

92 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 ...

**2**

votes

**1**answer

93 views

### What is the corank of a proper char subgroup of a finite index subgroup of a free group?

Let $F$ be the free group of rank $\aleph_0$, let $L \leq F$ be a finite index subgroup, and let $M$ be a proper characteristic subgroup of $L$. Is it possible that there exists a finite subset $S ...

**8**

votes

**1**answer

439 views

### Is there a simple description of this group?

I would like to know if there is a simple description of the following group. It has 2 generators whose the square of the commutator is trivial.
$$G=\langle a,b | (aba^{-1}b^{-1})^2=1\rangle$$
By ...

**1**

vote

**0**answers

48 views

### Question about Eigenvalues of group elements [migrated]

The quaternion algebra is given by
$\mathbb{H}$ = $\{a+bi+cj+dk \mid a, b, c, d \in \mathbb{R}, i^2 = j^2 = k^2 = -1, ij = k = -ji\} := \{z_1+z_2j \mid z_1, z_2 \in \mathbb{C}\}.$
I consider the ...

**5**

votes

**2**answers

183 views

### Condition(s) for the full autormophism group $\operatorname{Aut}(C(G, S))$ of the Cayley graph of $G$ to be isomorphic to $G$

If $\Gamma = C(G, S)$ is the (undirected) Cayley graph of a finite group $G$ with generating set $S$, then $G \le \operatorname{Aut}(\Gamma)$, the "full" automorphism group of $\Gamma$.
When is ...

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**1**answer

135 views

### Finite generation and profinite completion

Let $G$ be a (countable) residually finite group whose profinite completion is topologically finitely generated. Must $G$ be finitely generated?

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votes

**0**answers

152 views

### Generalization of the fundamental theorem of cyclic groups 2

This post is a sequel of Generalization of the fundamental theorem of cyclic groups
Let $G$ be a finite group then the fundamental theorem of cyclic groups can be formulated as follows:
Theorem: $G$ ...

**5**

votes

**1**answer

455 views

### An isomorphism between different Ext's coming from group cohomology

Let $G$ be an abelian group and $M$ a $G$-module with trivial action. It is well-known that $H^2(G,M)$ classifies extensions of $G$ by $M$, which is $\mathrm{Ext}^1_{Ab}(G,M)$.
On the other hand ...

**5**

votes

**1**answer

303 views

### Generalization of the fundamental theorem of cyclic groups

Let $G$ be a finite group then the fundamental theorem of cyclic groups can be formulated as follows:
Theorem: $G$ is cyclic iff it admits no two different subgroups with the same order.
proof: see ...

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votes

**0**answers

100 views

### What are the general zonal spherical functions for ${\rm SO}(n)/{\rm SO}(n-1)$?

The zonal spherical functions [1] on the sphere $(G={\rm SO}(n)$, $K={\rm SO}(n-1))$ are the Gegenbauer or ultraspherical polynomials if one considers the irreducible representations of ${\rm SO}(n)$ ...

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votes

**3**answers

1k views

### Is every abelian group a colimit of copies of Z?

More precisely, is every abelian group a colimit $\text{colim}_{j \in J} F(j)$ over a diagram $F : J \to \text{Ab}$ where each $F(j)$ is isomorphic to $\mathbb{Z}$?
Note that this does not follow ...

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votes

**1**answer

155 views

### Finite solvable groups are generated by a nilpotent subgroup + K elements?

Is there a constant $K \in \mathbb{N}$ such that for every finite solvable group $G$, there exists a nilpotent subgroup $N \leq G$, and a subset $S \subseteq G$ with $|S| \leq K$, and $\langle ...

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votes

**0**answers

95 views

### Reference for class of involutions containing longest element of finite Coxeter group?

Consider a finite (say irreducible) Coxeter group $W$ with a fixed generator set $S$ and rank $n$. This is the same thing as a finite real reflection group, generated by a set of “simple” ...

**5**

votes

**1**answer

169 views

### Kleinian groups containing an isomorphic copy of itself

Is there any example of a Kleinian group (acting on $\mathbb{H}^n$, $n \ge 3$) that contains a finite index isomorphic copy of itself? Here I don't consider Kleinian groups that only have parabolic ...

**-1**

votes

**1**answer

155 views

### Why do we not lose any generality by proving it only for finitely generated groups [closed]

In the proof of following theorem, in a paper by Farkas-
Here $\Delta(G) = \{ g \in G : |G:C_G(g)| < \infty \}$ and $U_1(\mathbb{Z}G) $ is the set of normalized units of the integral group ring ...