# Tagged Questions

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

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### A particular example of solvable group

Let be $m$ an integer, $m>0$, $n=p_1^{a_1}...p_t^{a_t}$. I define $f(n)=a_1+...+a_t$.
Let $G$ be a finite group and define $f(G)=max(f(|g|):g\in G)$.
Is there a finite solvable group $G$ with ...

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41 views

### Elements whose orders are multiple of $p$

Let $G$ be a non-solvable group, $N$ a cyclic normal subgroup of order $p$ with $p\notin \pi(G/N)$, $N=C_G(N)$ and $K=G/N\cong A_5$. BY these assumption we can conclude that $G$ has elements of orders ...

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611 views

### Does the symmetric group on an infinite set have a minimal generating set?

To clarify the terms in the question above:
The symmetric group Sym($\Omega$) on a set $\Omega$ consists of all bijections from $\Omega$ to $\Omega$ under composition of functions. A generating set ...

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**1**answer

133 views

### The special subgroups of a finite abelian group of rank two

Let $G=\langle a_{1}\rangle\times\langle a_{2}\rangle$ such that $|a_{i}|=2^{k_{i}}$ and $k_{1}>k_{2}$ and $H$ be a subgroup of $G$ that there exists an automorphism of $G$ such that fix only ...

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102 views

### Does $G\times H$ have a dual when $G$ and $H$ have?

Let $G$ and $H$ be two groups with duals. Does $G\times H$ have a dual?
A group $G$ has a dual iff the lattice of its subgroups is order-isomorphic to the dual of the subgroup lattice of some other ...

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385 views

### A double centralizing theorem for finite groups

I have a proof for the following assertion which employs Model Theory. It has certainly a pure group theoretic proof, but what is such a proof? Is the assertion trivial?
Theorem Let $G$ be a finite ...

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**1**answer

153 views

### Making a profinite group free

Let $F$ be a free profinite group, $G$ a profinite group. Suppose that the free profinite product $F \amalg G$ is a free profinite group. Must $G$ be a free profinite group?
For abstract groups the ...

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123 views

### Quantum oscillator and hidden symmetry [on hold]

Here I am trying to share my notes more clearly.
There are a number of works about hidden symmetry of quantum oscillator. To best of my knowledge the very first example of hidden symmetry was ...

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**3**answers

203 views

### Stallings' Theorem for free products of groups

There is a well known theorem which states that:
Theorem(Stallings):
For any immersion $f$ from a finite graph $D$ to $G$ there is a finite-sheeted covering space $D '$ of $G$ that extends $f$. ...

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54 views

### an upper bound for $K_G(G-N)$ [on hold]

Suppose $A$ is an arbitrary subset of group $G$ and $K_G(A)$ be the number of $G$-conjugacy classes contained in $A$. A finite group $G$ satisfies property $P_n$ if for every prime integer $p$, $G$ ...

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220 views

### groups of order $ p(p^2-1) / 4 $ where $p$ is a prime

Let $p> 3$ be a prime number and $G$ be a finite group of order $p(p^2-1) / 4 $. If $ 4 \mid (p+1)$ then easily we can see that the $ p$-Sylow subgroup of $G$ is a normal subgroup of $G$. As I ...

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199 views

### Union of conjugates of a closed subgroup of a compact group

Let $G$ be a compact Hausdorff group, $H \leq G$ a closed subgroup of infinite index in $G$.
Is it possible that the conjugates of $H$ cover some open neighbourhood of $1$ in $G$ (or the whole of ...

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114 views

### Representations of orthogonal groups vs representations of reflection groups

Let $V$ be a finite dimensional inner product space and $O(V)$ the orthogonal group of $V$.
Let $G$ be a (say, finite) reflection group on $V$, regarded as a subgroup of $O(V)$ ($G< O(V)$.) Let
us ...

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**1**answer

553 views

### solvable word problem without algorithm

Let $G$ be a finitely generated group. I wonder if there are examples where:
1) The word problem is known to be solvable in $G$ but there is no algorithm known.
2) The word problem is known to be ...

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169 views

### Union of conjugates of a subgroup

Let $G$ be a finite group, $H \leq G$ a proper subgroup. It is well known that the union of the conjugates of $H$ does not cover $G$. I would like to know of more precise results (even in special ...

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76 views

### What are the possible finite non-solvable quotients of one relator groups?

Is there a one-relator group with some finite non-solvable quotient, that does not have all large alternating groups as finite quotients?

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234 views

### Examples of fundamental domains

Everyone knows that it's difficult to compute a general fundamental domain for an arithmetic group but are there any specific examples where such domains have been calculated? I'm mostly interested in ...

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43 views

### The set of (property) elements of a locally compact group is closed

For which properties $(P)$ is the following statement known to be true?
In any locally compact group $G$, the elements of $G$ that satisfy $(P)$ form a closed subset of $G$. In other words, the ...

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**1**answer

434 views

### 1D TQFT in Freed-Hopkins-Lurie-Teleman

In the first section of Freed-Hopkins-Lurie-Teleman they construct a one-dimensional Topological Quantum Field Theory.
$F(\circ_+)$ is a vector space and $F(\circ_-)$ is the dual.
$F(\circ-\circ)$ ...

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92 views

### What is the importance of homological functor? [closed]

I am computing the homological functor of a group such as the nonabelian tensor square, nonabelian exterior square and etc. I am wondering what is the importance of computing this? What is their ...

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88 views

### Equidistribution of double coset

Let $G=PGL_n(\mathbb{R})$, $K=PO_n(\mathbb{R})$ and $X=G/K$. Also suppose $\Gamma=SL_n(\mathbb{Z})$ acts on the left of $X$. We define a typical Hecke operator on $L^2(\Gamma\backslash X)$ by the ...

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240 views

### A sequence of subsets of an infinite group

Is there an infinite group $G$ such that there is not any sequence $(A_n)$ of its subsets such that always
$$A_n=A_n^{-1}, \quad A_{n+1}A_{n+1}\subsetneqq A_n$$
?
link

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349 views

### Orders of Finite Simple Groups

Which finite simple groups have order N so that N+1 is a proper power?
As an example: the simple group of order $168=13^2-1$.

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131 views

### “Twisted” Lyndon equation in a free group

In 1959, Lyndon showed that in a free group, the equation $u^2v^2=w^2$ has only commuting solutions: $uv=vu=w$. Is there in the litterature any information about the following "twisted" version of the ...

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248 views

### Subgroup property stronger than being characteristic

In what follows, all groups are assumed to be finite.
Recall that if $K \leq H \leq G$ are groups, $K$ is said to be a weakly closed subgroup of $H$ in $G$ if, for all $g \in G$, $g^{-1}Kg \leq H$ ...

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18 views

### Improper Rotations in Even Dimensions [migrated]

In odd dimensions, we can represent any improper "rotation" G as $-\mathbb{1}\cdot R$ where $R\in SO(d)$. In even dimensions, $-\mathbb{1} \in SO(d)$ and we cant do this. Is there a way of writing an ...

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119 views

### Under what conditions there is a one-to-one mapping between a product of matrices and the sequence of matrices leading to the product?

I have a set of matrices $A_1,\ldots,A_n$. Let $\mathcal{A} = \{A_i\}$.
What are some simple conditions under which for any sequence of indices between $1$ and $n$, $a_1,\ldots,a_m$, the product
...

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146 views

### On the Suzuki group

Let $G$ be the Suzuki group over the field with $q=2^{2m+1}$ elements, $m>0$. Then, by Theorem 3.10 from B. Huppert, N. Blackburn, Finite Group III, pp 192-193, or wikipedia, the group $G$ contains ...

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151 views

### Switching representatives of right coset in a paper on fundamental domain of tree of GL(2)

I have a question concerning the following paper: "The fundamental domain of the tree of GL(2) over the function field of an elliptic curve" by Shuzo Takahashi. (1993, Duke Math. J., Vol. 72, No. 1). ...

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62 views

### Intersection of Maximal Left Ideals with Finite Dimensional Quotient

Let $\Gamma$ be a finitely generated group and let $A=\mathbb{C}[\Gamma]$ be the corresponding group algebra over $\mathbb{C}$. Let $X$ be the set of all maximal left ideals of $A$ and let $X_0=\{I ...

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### Fixed points of $x\mapsto 2^{2^{2^{2^x}}} \mod p$

Let $p$ be a prime. For how many elements $x$ of $\{0,1,\dotsc,p-1\}$ can it be the case that
$$2^{2^{2^{2^x}}} = x \mod p?$$
In particular, can you find a simple proof (or, even better, several ...

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80 views

### Acylindrical hyperbolic groups

In Osin's paper "Acylindrically hyperbolic groups" in Lemma 5.7, there is a condition, that $|X_1\triangle X_2|<\infty$ for two relative generating sets. I'm sorry, but I didn't find a definition ...

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123 views

### Semi direct product group

Suppose $G=V \rtimes M$, is a semi product group of an elementary abelian p-group of size $|V|=p^e$ and $M$
is a subgroup of $G$. If $f$ is the natural projection from $G$ onto $M$.
$C_x=\{x^G\}$ is ...

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**1**answer

160 views

### Bases of free groups

Let $F$ be a free group on a finite set $X$. Let $A \subseteq X$ be a subset of $X$ contained in some $H \leq F$, a subgroup of finite index in $F$. Must there be a basis (free generating set) for $H$ ...

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169 views

### Kaplansky's unit conjecture and unique products

There are three conjectures on group rings that bear the name of Kaplansky (see for example this question). The 'unit conjecture' in the title of the present question is the strongest of them, and ...

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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,~~~~~ ...

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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
...

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592 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 $L(n)$ ...

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188 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 ...

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62 views

### Order of elements in a solvable group [duplicate]

Let be $n$ an integer, $n>0$, $n=p_1^{a_1}...p_t^{a_t}$. 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 wonder if it is true:
if $G$ is ...

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**1**answer

363 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 ...

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133 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 ...

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134 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 ...

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**1**answer

89 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 ...

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114 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 ...

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**1**answer

860 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 ...

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600 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.
...

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**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 ...

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87 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 ...

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**1**answer

188 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 ...