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

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63 views

### Long cycles in $\text{Sym}(n)$ [on hold]

Let $x,y \in \text{Sym}(n)$ (symmetric group on $\{ 1,2,\ldots,n \}$) and $z:=xy$.
Question: What non-trivial sufficient conditions (on $x$ and $y$) for $z$ to be a cycle of length $n$ do we know?
...

**-3**

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**0**answers

66 views

### Is there a group-theoretic proof of the Riemann rearrangement theorem? [on hold]

The analytic proofs are easy to understand but they don't explain why commutativity breaks down when you go from finite sums to infinite sums of real numbers. I suspect that this involves the action ...

**0**

votes

**0**answers

63 views

### Bases and transversals

Let $F$ be a free finitely generated group, $L \leq H \leq F$ subgroups of finite index.
Given bases $B$ of $F$ and $C$ of $H$, must there be a Schreier transversal $T$ for $H$ in $F$ such that the ...

**2**

votes

**1**answer

76 views

### Isolated elements of primary order ($Z^*$-theorem revisited)

Let $G$ be a finite group, $p$ a prime, $P\in{\rm Syl}_p(G)$, and $x\in P$.
Let $Z^*_p(G)$ denote the full preimage in $G$ of $Z(G/O_{p'}(G))$ under the canonical epimorphism $G\to G/O_{p'}(G)$.
...

**2**

votes

**2**answers

151 views

### Bound on the index of an abelian subgroup in discrete subgroup of the euclidean group?

I asked the following question on math.SE a couple of days ago. Dietrich Burde gave an answer for the case that the subgroup is not only discrete but also acts cocompactly.
What about the general ...

**3**

votes

**1**answer

214 views

### Generating finite groups

Let $G$ be a finite group possessing a generating set of order $n \in \mathbb{N}$. Let $H \leq G$ and $x_1, \dots, x_n \in G$ for which $\langle H, x_1, \dots, x_n \rangle = G$. Must there be $h_1, ...

**4**

votes

**1**answer

130 views

### Orbit spaces of crystallographic groups

In their paper "On Three-Dimensional Space Groups", Conway et al. write
Although this paper was inspired by the orbifold concept, we did not need to consider the 219 orbifolds of space groups ...

**13**

votes

**0**answers

439 views

### What's the big deal about $M_{13}$?

$M_{13}$ is the Mathieu groupoid defined by Conway in
Conway, J. H. $M_{13}$. Surveys in combinatorics, 1997 (London), 1–11,
London Math. Soc. Lecture Note Ser., 241, Cambridge Univ. Press, ...

**1**

vote

**1**answer

130 views

### Unipotent conjugacy classes

Consider a connected reductive group G over the complex numbers. Is there a `simple' formula for the number of conjugacy classes of unipotent elements in G?

**5**

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**0**answers

74 views

### representable functions on FinGrp

We look at functions $f: \text{FinGrp} \rightarrow \mathbb{N}$ such that $f$ is constant on isomorphism classes.
Let's say that $f$ is representable if there is a (possibly infinite) group $K$ such ...

**0**

votes

**0**answers

93 views

### Name/terminology for a relationship between group actions

Let $G$ and $H$ be groups, both acting on a set $X$. Suppose that there is a homomorphism $\phi:G\to H$ such that for every $g\in G$ and $x\in X$, $g\cdot x = \phi(g)\cdot x$. Is there a name for this ...

**2**

votes

**0**answers

47 views

### Separation of topological group elements by invariant neighbourhooods

Let $G$ be a topological group that is Hausdorff, that is, for every pair $(g,h)$ of distinct elements of $G$, there exist disjoint open sets $U_g$ and $U_h$ such that $g \in U_g$ and $h \in U_h$.
...

**3**

votes

**0**answers

99 views

### Parshin's buildings for higher local fields

What is the status of the theory of buildings for higher local fields?
I know that there are some papers of Parshin, in which he describes some examples, like $PGL_2$ and $PGL_3$ over ...

**2**

votes

**0**answers

130 views

### On groups satisfying a law

We say that a group $G$ satisfies a law if there exists a (nontrivial) word $w \in \mathbb{F}_n$ such that $w(g_1,\dots,g_n)=1$ for every $g_1,\dots, g_n \in G$. For example, any abelian group ...

**0**

votes

**1**answer

186 views

### Where can I find the classification of groups of order 16p? [closed]

I need to classify the groups of order $16p$ by their generators and relations between the generators. Can I find this classification anywhere?

**0**

votes

**2**answers

220 views

### Goldbach-type problem: the valuation of irreducible elements of a subgroup of $\mathbb{Q}_{> 0}$

Let $\phi : \mathbb{Q}_{>0} \to \mathbb{Z}$ be the group morphism defined by $\phi(p) = p$ for $p$ a prime number.
It follows that $\phi(\prod_i p_i^{n_i}) = \sum_i n_i p_i$, with $p_i$ a prime ...

**1**

vote

**1**answer

116 views

### Products of subgroups that generate a finite group

Consider the following general problem. There is a finite group $G$ and $H_1,H_2 < G$. Suppose we know that $\langle H_1, H_2 \rangle = G$, i.e. $G$ is generated by $H_1$ and $H_2$. Denote by $n_0$ ...

**5**

votes

**0**answers

199 views

### A particular example of solvable group

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

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votes

**2**answers

783 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 ...

**2**

votes

**1**answer

153 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 ...

**1**

vote

**0**answers

108 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 ...

**3**

votes

**2**answers

461 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 ...

**2**

votes

**1**answer

170 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 ...

**3**

votes

**3**answers

213 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 $f$. ...

**3**

votes

**2**answers

277 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 ...

**6**

votes

**2**answers

213 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 ...

**7**

votes

**0**answers

127 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 ...

**4**

votes

**1**answer

572 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 ...

**6**

votes

**3**answers

267 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 ...

**5**

votes

**0**answers

80 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?

**7**

votes

**1**answer

244 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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**0**answers

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 ...

**2**

votes

**1**answer

455 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.
$F(\circ-\circ)$ ...

**3**

votes

**0**answers

89 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 ...

**9**

votes

**1**answer

254 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

**3**

votes

**1**answer

359 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 $168=13^2-1$.

**6**

votes

**0**answers

139 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 ...

**6**

votes

**2**answers

257 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 $g \in G$, $g^{-1}Kg \leq H$ ...

**-1**

votes

**0**answers

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 ...

**0**

votes

**1**answer

120 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
...

**2**

votes

**1**answer

147 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 ...

**3**

votes

**1**answer

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). ...

**3**

votes

**1**answer

64 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 ...

**30**

votes

**3**answers

2k views

### 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 ...

**2**

votes

**0**answers

85 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 ...

**0**

votes

**1**answer

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 ...

**2**

votes

**1**answer

161 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 $H$ ...

**16**

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184 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 ...

**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
...