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

**2**

votes

**0**answers

105 views

### Subsets of a group as an algebraic structure

Let $F$ be a set and $.$ be a binary operation on $F$ and $.^{-1}:F\to F$ be a so-called inverse operation on $F$ such that $(F,.)$ is semigroup and for each $x,y\in F$,
$$(x^{-1})^{-1}=x,~~~~~ ...

**-2**

votes

**0**answers

54 views

### examples of polyclic groups [migrated]

From the notes Coarse diﬀerentiation and the geometry of
polycyclic groups, I found a theorem
$\Gamma$ is polycyclic iff $\Gamma$ is a lattice in a solvable
unimodular lie group $G$ - Mostow
...

**18**

votes

**3**answers

598 views

### Number of primitive $n$th roots with positive versus negative real parts

Does anyone know a reference to the following results, which I can prove, but I suspect may be known. Let $R(n)$ denote the number of primitive $n$th roots of unity with positive real part, and $L(n)$ ...

**7**

votes

**0**answers

194 views

### Normal generators of finite index subgroups in a free group

Let $F=F(a,b)$ be the free group of rank $2$.
Question 1: Given any positive integer $d$, can one always find elements $u_j,v_j,w_j \in F$, $j=1,\dots,d$, such that if $1 \le j <k \le d$ then the ...

**14**

votes

**1**answer

366 views

### Permutation Groups Containing non-commuting $p$-cycles

I noticed that the following is true, and that there is a reasonably elementary proof of it (in particular, the classification of finite simple groups is not needed). Let $G$ be a finite permutation ...

**3**

votes

**1**answer

134 views

### Is it a Frobenius complement?

Let $n$, $m\in \mathbb{N}$. Let $p$, $q$ be primes with $q^{n}|p-1$. Let $H$ the semidirect product of a cyclic group $A=C_{p^{\large{m}}}$ by a cyclic group $B=C_{q^{\large{n+1}}}$ which induces an ...

**4**

votes

**1**answer

135 views

### Generators of Sylow subgroups

Is there a function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for each finite supersolvable group $G$, and a Sylow subgroup $S \leq G$ we have $d(S) \leq f(d(G))$?
Here $d(H)$ denotes the ...

**-1**

votes

**1**answer

93 views

### Relation of the order of elements in a metabelian group [closed]

Let $n$ be an integer, $n>1$, $n=p_1^{a_1}...p_t^{a_t}$. I define $f(n)=a_1+...+a_t$.
Let be $G$ a finite group and I define $f(G)=max(f(|g|):g\in G)$.
I have to prove that if $G$ is a ...

**2**

votes

**0**answers

116 views

### Dehn functions of Thompson's group $F$

It's well know that the first order Dehn function of $F$ is quadratic. Is a similar result known for its second-order, or even higher-order, Dehn function?
The second-order Dehn function of a group ...

**23**

votes

**1**answer

865 views

### Enumeration of a finite group

Let $G=\{g_1,g_2,...,g_n\}$ be a group with $e=g_1$ and $n$ is odd,
Set $$a_1=g_1$$
$$a_2=g_1g_2$$
$$a_3=g_1g_2g_3$$
$$a_n=g_1g_2...g_n$$
I am looking for example that all $a_i$ are different from ...

**14**

votes

**0**answers

646 views

### Is there an infinite field of characteristic 2 whose multiplicative group is torsion free and (direct-sum) indecomposable?

Let $F$ be a infinite field of characteristic 2 whose multiplicative group $F^*$ is torsion free. I would like to conclude that $F^*$ is decomposable or find an example where $F^*$ is indecomposable.
...

**1**

vote

**1**answer

81 views

### Number of non isomorphic groups in Cext(G,C_p)

I want to modify my question mathoverflow.net/questions/50922. Let $C_{p^e}$ be a cyclic group of order $p^e$, $p$ prime. Denote by $\text{Cext}(G,C_p)$ the group of all central extensions of $C_p$ by ...

**6**

votes

**0**answers

89 views

### Two-relator products of cyclic groups

In "A proof of the Scott–Wiegold conjecture on free products of cyclic groups" Howie proved that every one-relator product of three cyclic groups is nontrivial. Is there a now proven theorem that says ...

**3**

votes

**1**answer

189 views

### How does associativity get twisted by elements of $H^3(G)$?

In Braided Monoidal Categories by Joyal and Street, §6 a monoidal category is $V =T(G,M,h)$ built using a recipe:
objects are are elements of $G$ ✓
$V_0(x,y) = M$ if $( x=y)$ or else ...

**2**

votes

**0**answers

90 views

### Orbit-Stabilizer theorem for continuous groups

The orbit-stabilizer relationship (also known as the orbit-stabilizer theorem) is very clear for finite groups. Is there an equivalent relation for continuous groups?
Also, is there a similar notion ...

**9**

votes

**1**answer

152 views

### Fixed set of order p automorphism of Bruhat-Tits tree

I would like to know the structure of the fixed set of an order $p$ automorphism [Edit: induced by a matrix in $GL_2(K)$] on the Bruhat-Tits tree for a p-adic field $K$, specifically in the case where ...

**11**

votes

**1**answer

392 views

### On the order of finite simple groups

About the order of finite simple groups there exists a very interesting result which stated as follows:
Let $G$ be an non-solvable simple group of order $g$. If $p\mid g$, where $p>g^{1\over 3}$ ...

**11**

votes

**0**answers

218 views

### Are there additive subgroups of reals of dimension 1 with no subgroups of dimension strictly between 0 and 1?

I will use $dimA$ to denote the Hausdorff dimension of a set $A \subseteq \mathbb{R}$. Being a null set means having Lebesgue measure zero.
In the 1966 paper "Additive gruppen mit vorgegebener ...

**1**

vote

**2**answers

240 views

### A semisimple group ring

Let $n \in \mathbb{N}$, $p$ a prime number, and $G$ a finite group of order coprime to $p$. Let $R = \mathbb{Z} /p^n \mathbb{Z}$ be the ring of integers mod $p^n$. Must $R[G]$ be semisimple?
As noted ...

**4**

votes

**0**answers

103 views

### Does this class of groups contain finitely generated infinite periodic groups?

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 ...

**0**

votes

**1**answer

99 views

### Direct limit of primitive integral matrices

I'm interested in computing the direct limit of an arbitrary $2\times 2$ primitive matrix over $\mathbf{Z}$. That is for a fixed primitive matrix $M$, the colimit ...

**5**

votes

**1**answer

216 views

### Index of congruence modular subgroup of level (1,d)

Let $D = \text{diag}(1,d)\in M_{2}(\mathbb{Z})$ be a $2\times 2$ matrix, where $d$ is an odd integer. We define the subgroup $\Gamma_D\subset M_{4}(\mathbb{Z})$ as:
$$\Gamma_D := \left\lbrace R\in ...

**0**

votes

**0**answers

77 views

### Actions and representations of profinite groups

Let $p$ be a prime number, and denote by $\mathbb{Z}_p$ the additive profinite group of p-adic integers. Let $G$ be a finitely generated profinite group of order coprime to $p$, and $V = ...

**5**

votes

**2**answers

360 views

### How to estimate the Haar measure on $G_2$

I have a sequence of real numbers. I want to know whether this sequence looks like the traces in the standard representation of a random sequence of elements of $G_2$. (Here random is according to the ...

**4**

votes

**2**answers

314 views

### Finite groups factorized into two simple alternating groups

My research is somehow related to the following question :
Describe and classify all finite groups $G$ such that $G=HK$ with $H \cap K=1$, where $H \cong A_m$ and $K \cong A_n$ for some integers $m, ...

**3**

votes

**1**answer

148 views

### Integral representations of groups of small order

I have a problem in which it would be helpful to know about the integral representations of some groups of small order (probably of fairly low degree). From what I've gathered so far, cyclic groups of ...

**5**

votes

**1**answer

382 views

### A generalized Burnside's lemma

Let $G$ be a finite group acting on a set $X$, and let $S\subseteq G$ be a union of conjugacy classes. Then I believe I can prove:
$$ \sum_{[x]\in X/G} \frac{|G_x \cap S|}{|G_x|} = \sum_{g\in S} ...

**2**

votes

**0**answers

65 views

### Semidirect products of semigroups [closed]

Let $S,T$ be two semigroups. A function $f:S\to T$ is called multiplicative if for any $x,y\in S$ we have $f(xy)=f(x)f(y)$. If $T=S$ then $f:S\to S$ is called automorphism on $S$.
A function ...

**0**

votes

**0**answers

117 views

### classify antiholomorphic involutions of projective space

On $\mathbb{CP}^1$ with standard complex structure, how to prove that there are only two types of antiholomorphic involution, given by
$$ \tau :[z:w]\mapsto [\bar w, \bar z] \qquad \eta :[z:w]\mapsto ...

**7**

votes

**2**answers

395 views

### Elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$

For some research work, I need to know the classification of elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$, up to conjugation.
Since I essentially need $n\le 4$, I think that I can show it ...

**2**

votes

**2**answers

149 views

### Number of generators of the commutator

Can one find a function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for every finite supersolvable group $G$ we have: $d(G') \leq f(d(G))$?
Here $d(K)$ is the cardinality of a minimal set of ...

**1**

vote

**1**answer

268 views

### Kropholler's Conjecture and 3-manifolds

Suppose that $G$ is a group and that $H$ is a subgroup, both finitely generated, and assume that there is a non-trivial H-almost invariant set $X$ with $HXH=X$. Kropholler's Conjecture asserts that $ ...

**10**

votes

**1**answer

326 views

### Categorical interpretation of disjoint cycle notation for tracing permutations

For a natural number $n\in\mathbb{N}$, let $\underline{n}$ denote the finite set $$\underline{n}:=\{1,2,\ldots,n\}.$$
A permutation $\sigma\in Aut(\underline{n})$ can be uniquely written (up to order) ...

**1**

vote

**1**answer

168 views

### Countable residually finite abelian groups

Is there a countable abelian group for which its subgroups are exactly all of the countable "reduced" abelian groups? (Reduced means that its divisible subgroup is zero)
Is the following group a ...

**5**

votes

**0**answers

162 views

### Is translation by the free group (in two generators) on a certain completion of the group an amenable action?

Let $\mathbb{F}_2 = \langle a,b\rangle$ be the free group in two generators $a,b$ and let $\alpha \in \text{End}(\mathbb{F}_2)$ be given by $\alpha(a) = a^2, \alpha(b)= b^2$. Note that the index ...

**0**

votes

**0**answers

125 views

### Trivial extensions by torsion-free groups

Let $A$ be an abelian group. Recall that $A$ is
($\bullet$) a Whitehead group if $\text{Ext}(A,\mathbb Z)=0$,
($\bullet$) a free abelian group if $\text{Ext}(A,D)=0$ for every abelian group $D$.
...

**4**

votes

**0**answers

73 views

### Generators of the symplectic subgroup $\Gamma^g(1,2)$

Let $\mathbb{A}^{m\times n}$ denote the set of all $m \times n$ matrices with entries in the set $\mathbb{A}$. For a matrix $M$ we let ${^tM}$ denote its transpose, and $M^{-1}$ its inverse, if it is ...

**19**

votes

**0**answers

646 views

### Non-linear expanders?

Recall that a family of graphs (indexed by an infinite set, such as the primes, say) is called an expander family if there is a $\delta>0$ such that, on every graph in the family, the discrete ...

**1**

vote

**1**answer

98 views

### Modules “projective in a subcategory”

In my research I have come up with the following notion which I would like to learn more about. It may be very naive.
Let $R$ be a ring, $M$ an $R$-module and $S$ a class or $R$-modules closed under ...

**3**

votes

**2**answers

145 views

### Elementary abelian $p$-subgroups of maximal rank in finite groups of Lie type

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $G$ be a reductive group defined over $\mathbb{F}_p$. For any $d\in\mathbb{Z}^+$, let $C_d(G)$ be the set of conjugacy ...

**5**

votes

**0**answers

83 views

### Tensor product of dual groups

Let $G,H$ be compact abelian groups, $G^*,H^*$ be their Pontryagin duals, $G^*\otimes H^*$ the tensor product of $G^*,H^*$ and $K=(G^*\otimes H^*)^*$. Does the group $K$ have a special name? What is ...

**4**

votes

**2**answers

196 views

### Maximal order of finite subgroups of $GL(n,Z)$

I am interested in the finite subgroups of $GL(n,Z)$ of maximal order.
Except for the dimensions $n = 2,4,6,7,8,9,10$ they are -- up to conjugacy in $GL(n,Q)$ -- in each dimension the group of signed ...

**1**

vote

**1**answer

110 views

### the lower bounded of metrics on a group

Let $G$ be a finitely generated group, S is a set of generators. If $\forall s\in S, n\in \mathbb{Z}$, $\exists C>0$ such that $|s^n|_S\ge C|n|$, does it imply $\forall$ infinite order element ...

**0**

votes

**1**answer

99 views

### Amenability of the Koopman representation

Let $G$ be a locally compact group which acts non-singularly on a standard probability space $(X,\mu)$. Consider the Koopman representation $\pi_X:G\rightarrow U(L^2(X,\mu))$ defined by
...

**6**

votes

**1**answer

267 views

### Is there any criterion for whether the group G is isomorphic to the automorphism group of another group?

I was reading This question on MO and suddenly faced another question.
Is there any necessary and sufficient condition for $G$ such that there exist a group $G'$ that $G \cong Aut(G')$ ?
What can we ...

**8**

votes

**2**answers

423 views

### When is the conjugation character almost multiplicity free?

Let me give some motivation, which also explains how I arrived at the question. We may let the finite group $G$ act on itself by conjugation, and this makes the group ring into a $\mathbb{Z}G$-module ...

**7**

votes

**0**answers

170 views

### A strong sum-product “for translates” in finite fields

In the course of some recent research, I've sketched out a proof of the following result. My basis question is: is the result interesting?
Proposition There exists an absolute constant $c$ such ...

**7**

votes

**0**answers

143 views

### Geometric characterization of property (T) and the Haagerup property

Are there geometric characterizations of property (T) and the Haagerup property, i.e., such that can be stated in terms of a Cayley graph, in analogy to the Folner set characterization of amenability? ...

**28**

votes

**1**answer

539 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 ...

**1**

vote

**0**answers

154 views

### A question on the poset of classes of isomorphic subgroups of finite groups

Given a finite group $G$, we consider the set $${\rm Iso}(G)=\{[H]\mid H\leq G\},$$where
$[H]=\{K\leq G\mid K\cong H\}, \forall H\leq G$. Then ${\rm Iso}(G)$ can be partially ordered by defining
...