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

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2
votes
1answer
76 views

Double quotients of Coxeter groups have the chain property?

Let $(W,S)$ be a Coxeter group with Bruhat order $\leq$ and length function $\ell(w)$. Definition: a subset $X \subset W$ has the chain property if whenever $x,y \in X$ with $x < y$, there ...
0
votes
0answers
78 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
1answer
104 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
2answers
170 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
1answer
234 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, ...
5
votes
1answer
149 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
0answers
460 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
1answer
137 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
votes
0answers
79 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 ...
1
vote
0answers
99 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 ...
3
votes
0answers
49 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
0answers
108 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
0answers
135 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
1answer
189 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
2answers
226 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
1answer
126 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
0answers
205 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 ...
23
votes
2answers
808 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
1answer
176 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
0answers
114 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
2answers
464 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
1answer
178 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
3answers
242 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
2answers
285 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
2answers
215 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 ...
8
votes
0answers
134 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
1answer
582 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
3answers
296 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
0answers
91 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
1answer
253 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 ...
0
votes
0answers
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
1answer
466 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
0answers
92 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 ...
10
votes
1answer
264 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
1answer
368 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
0answers
143 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
2answers
260 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 $g \in G$, $g^{-1}Kg \leq H$ ...
0
votes
1answer
160 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
1answer
151 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
1answer
153 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
1answer
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
3answers
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
0answers
93 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
1answer
127 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
1answer
169 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
votes
0answers
207 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
0answers
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,~~~~~ ...
18
votes
3answers
607 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
0answers
203 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
1answer
374 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 ...