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

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31
votes
1answer
561 views

Identities of commutators

Let $G$ be a group and set $[x,y]:= x^{-1}y^{-1}xy$ as usual and consider it as a binary operation. Question: Is there a description of the identities that the operation $[.,.]$ satisfies for all ...
1
vote
0answers
76 views

Infinite Hirsch length [closed]

Can a residually finite group $G \in LFin \rtimes VPoly$ ($G$ is a semi-direct product of a locally finite group by a virtually polycyclic group) have an infinite Hirsch length?
2
votes
0answers
177 views

Mixed up by definitions of mildly mixing

Here are two setup where the notion of "mildly mixing" comes up: for representations and for group acting by measure preserving transformations (see definitions below). Since a natural class of ...
9
votes
1answer
209 views

p-groups such that the center is contained in many cyclic subgroups

I'm looking for examples of $p$-groups $G$ with the following three properties: the center of $G$ is $\mathbb{Z}/p$, and $G^{\text{ab}} = (\mathbb{Z}/p)^n$ for some $n$, and for every $g \in G$ ...
0
votes
0answers
67 views

Matrix representation of the Heisenberg quaternionic group

Equiped with the law $(a,b,c)\circ (a',b',c') = (a + a', b + b', c + c' + ab'),$ the matrix representation of the Heisenberg group $H^3= \mathbb C\times \mathbb R$ is given by $$ \begin{pmatrix} 1 &...
16
votes
3answers
705 views

Simplicity of $A_n.$

I am teaching an introductory group theory course, and it has come to the inevitable proof that $A_n$ is simple for $n\geq 5.$ Now, there seem to be a number of proofs that I can find - one the "...
0
votes
0answers
107 views

On the structure of groups according to their conjugacy classes

A finite group $G$ satisfies property $P_n$ if for every prime integer $p$, $G$ has at most $(n-1)$ non-central conjugacy classes the order of the representative element of which is a multiple of $p$. ...
3
votes
0answers
118 views

Cocycle condition for 2-groups

I know that if $\omega_d(g_1, \ldots, g_d)$ is an d-cocycle characterized by $H^d(G,U(1))$, it satisfies the co-cycle condition $(d\omega_d)(g_1, \ldots, g_{d+1}) = g_1.\omega_d(g_2,\ldots,g_{d+1}) + ...
1
vote
0answers
92 views

Exponent of the quotient of the commutator of a free group

Let $F$ be a free group on two generators, let $F^p$ denote the normal subgroup of $F$ generated by the $p$-th power, $p$ a prime. Do we know anything about the exponent of $[F,F]/[F^p,F]$. Edit: $G=...
5
votes
0answers
106 views

Mal'cev completions of finitely generated torsion-free nilpotent groups

There is some question from geometric group theory: One wonders if the following conditions are equivalent for finitely generated torsion-free nilpotent groups $\Gamma$ and $\Lambda$: $\Gamma$ and $\...
3
votes
0answers
67 views

Have locally principal crossed homomorphisms been studied?

Take a (multiplicative finite) group $H$ acting on the left (by automorphisms) on an (additive finite) abelian group $A$, and recall that the abelian (additive) group of crossed homomorphisms from $H$ ...
8
votes
1answer
215 views

Need a good name for an algorithmic problem in groups that generalizes the conjugacy problem

I am looking for a good name for the following problem: Given elements $g_1,\dotsc,g_n$ in a (finitely generated) group $G$, determine if the product of their conjugacy classes $g_1^G\dotsb g_n^G$ ...
1
vote
1answer
66 views

Character kernels in the lattice of subgroups of a finite abelian group

I am looking for any efforts that have been made to characterize the character kernels (equivalently, the subgroups yielding cyclic quotients) inside the lattice of subgroups of a finite abelian group....
1
vote
0answers
189 views

Semidirect product of semidirect products

For algebraic objects, say for groups $ N,K$ and algebra $A$, if we have the semidirect product of a semidirect product, $(A \rtimes_{\gamma} N ) \rtimes_{\theta} K$, are there conditions that would ...
0
votes
0answers
105 views

generalized word problem

If $H$ is a finitely generated subgroup of $G$ and if $H$ given by say a finite set of words which generate it, then the generalized word problem for $H$ in $G$ is the problem of deciding for an ...
7
votes
1answer
256 views

What does the unique mean on weakly almost periodic functions look like?

There is a unique invariant mean $m$ on WAP functions on any discrete group (see definitions below, theorem of ?). However, the proofs I found are fairly non-explicit on how to obtain this invariant ...
7
votes
1answer
307 views

Is $[729,57]$ a Sylow $3$-subgroup of some well-known group?

Let $G$ be the group $[729,57]$, using GAP's notation. I have so far two descriptions of the group: a presentation an embedding (not surjective!) of the group into a Sylow $3$-subgroup of the unit ...
1
vote
1answer
193 views

Finite groups $G$ satisfying property $P_n$

A finite group $G$ satisfies property $P_n$ if for every prime integer $p$, $G$ has at most $(n−1)$ non-central conjugacy classes the order of the representative element of which is a multiple of $p$. ...
1
vote
1answer
102 views

'Accidental' isomorphisms for $Spin^C(n)$

The complex spin groups $Spin^C(n)$ appear in the fibration $Spin(n)\rightarrow Spin^C(n)\rightarrow\ S^1$ which must split since $BSpin(n)$ is 3-connected to give a homotopy equivalence $Spin^C(n)\...
2
votes
0answers
74 views

Groups with special character degrees

Let $G$ be a finite group of order $p_1^{a_1}\times p_2^{a_2}\times\cdots \times p_n^{a_n}$. Is there any classification for simple groups such that for each $i$, $p_i^{a_i}$ is an irreducible ...
1
vote
2answers
218 views

Nilpotency of Lie Algebra from Structure Constants

Suppose we have a Lie algebra with structure constants $$\mathrm{d}e^i=\sum_{j<k}a_{ijk}e^j\wedge e^k$$ for some coefficients $a_{ijk}$. In this setting, how may be checked (perhaps ...
3
votes
1answer
176 views

Representations of p-groups where 1 is never an eigenvalue

Fix some $n \geq 1$ and some prime $p$. I'm looking for finite $p$-groups $G$ and finite-dimensional complex representations $V$ of $G$ with the following two properties: The abelianization of $G$ ...
12
votes
2answers
262 views

Are finitely generated amenable groups positively finitely generated?

Let $G$ be a finitely generated amenable group. Is there a positive integer $n$ such that $n$ random elements of $G$ generate it with positive probability? Being more formal, note that $G^n$ is ...
7
votes
3answers
464 views

Which groups are Galois over some p-adic field?

Suppose I have some finite $p$-group $G$, or a little extension of it. How do I know if there exists a prime $l$ and a finite extension $K$ of $\mathbb{Q}_l$ such that $G$ is the Galois group of ...
3
votes
1answer
188 views

Symmetry Group of a Polynomial

Given a polynomial $P \in \mathbb{Z}[X_1,\ldots,X_n]$, is there a poly-time algorithm which computes the group of permutations of variables that leaves $P$ unchanged? (Clearly, the trivial $O(n!)$-...
5
votes
1answer
185 views

Hyperoctahedral group acting on a special permutation

Let $[n]=\{1,...,n\}$ and $[\hat n]=\{\hat 1,...,\hat n\}$. Realize the hyperoctahedral group $H_n$ as the centralizer of the permutation $(1\hat 1)\cdots (n \hat n)$. It has $2^n n!$ elements. Let $...
6
votes
1answer
562 views

Positivity of the alternating sum of indices for boolean interval of finite groups

Let $G$ be a finite group and $H$ a subgroup such that the interval $[H,G]$ is a boolean lattice. Let $L_1, \dots , L_n$ be the maximal subgroups of $G$ containing $H$. Let the alternative sum ...
5
votes
1answer
109 views

A sum over characters of $S_{2n}$ and zonal spherical functions of $(S_{2n},H_n)$

The hyperoctahedral group $H_n$ can be seen as the centralizer of the permutation $(12)(34)\cdots (2n-1\,2n)$ in $S_{2n}$. It has $2^nn!$ elements. The quantities $$ \omega_\lambda(\pi)=\frac{1}{2^...
2
votes
0answers
87 views

Centralizer of a dense subgroup in a maximal subgroup of a reductive group

I am looking for a reference to the following statement "Let $G$ be a reductive algebraic group and $K$ a maximal compact subgroup of $G$. If $H$ is a dense subgroup in $K$, then the centralizer of $H$...
0
votes
1answer
294 views

When are groups subgroups of a same group?

I put this question on Stackexchange : http://math.stackexchange.com/questions/1659760/when-are-groups-subgroups-of-a-same-group but it got no answer, so I post it here. Let $\mathcal{G}$ be a ...
13
votes
2answers
669 views

Torsion-freeness of two groups with 2 generators and 3 relators and Kaplansky Zero Divisor Conjecture

Let $G_1$ and $G_2$ be the groups with the following presentations: $$G_1=\langle a,b \;|\; (ab)^2=a^{-1}ba^{-1}, (a^{-1}ba^{-1})^2=b^{-2}a, (ba^{-1})^2=a^{-2}b^2 \rangle,$$ $$G_2=\langle a,b \;|\; ...
1
vote
0answers
133 views

the braid groups on the sphere

The braid group $B_n(S^2)$ on the sphere possesses a finite element of order which generates a cyclic group $M$ in $B_n(S^2)$. My question is this: What is the index of $H$ in $B_n (S^2)$? Same ...
2
votes
0answers
84 views

Dicks–Dunwoody almost stability theorem

In the book 'Groups acting on graphs' (1989), Dicks and Dunwoody prove the following theorem (paraphrased): Let $G$ be a group acting on a set $E$, let $E'$ be a subset of $E$ and let $V$ be the set ...
1
vote
1answer
103 views

Can the reversed lattice of a subgroups interval be represented?

Let $G$ be a finite group and $H$ a subgroup. The interval $[H,G]$ is the lattice of overgroups of $H$. It is an open problem to know if every finite lattice can be represented by such an interval (...
3
votes
0answers
61 views

Infinite finitely generated groups whose Frattini factors are Klein 4-group

Is there an infinite finitely generated group whose Frattini factor is isomorphic to Klein 4-group?
4
votes
1answer
308 views

Maximal set of non commuting elements in a conjugacy class of $S_8$

Can someone calculate by computer or prove the subset of maximal order of pairwise non commuting elements in the set of conjugacy class containing $(123)(45)$ in $S_8$? I mean a subset of conjugacy ...
-1
votes
1answer
121 views

Can someone explain how Sims's algorithm works on a permutation group with a simple example? [closed]

Can someone explain how Sims's algorithm works on a permutation group with a simple example? The book "Permutation group algorithms" by Seress is a pretty hard read with a whole bunch of confusing ...
2
votes
1answer
252 views

A presentation for $GL(2,\mathbb{Z}/p^n \mathbb{Z})$

In 'A presentation of $PGL(2,p)$ with three defining relations' by E.F.Robertson and P.D.Williams, we can find a presentation of $PGL(2,p)$: $\langle a,b | a^2 = b^p = (a b^2 a b^r)^2 = (abab^r)^3 = ...
0
votes
1answer
86 views

Uniform pro-p groups as a semi-direct product

Let $G$ a finitely generated uniform pro-$p$ group. Then $G/[G,G]$ is abelian and so it is of the form $\mathbb{Z}_p^r\times T$ for some integer $r$ and finite $p$-group $T$. Therefore, $[G,G]$ is ...
3
votes
3answers
669 views

Square of non-zero element in group algebra is always non-zero?

Consider a finite group $G$ and complex group algebra $\mathbb{C}(G)$, i.e. formal sums $$ \sum_{g \in G} a_gg, \ a_g \in \mathbb{C},$$ with algebra structure: $$ \sum_g a_gg+\sum_gb_gg=\sum_g(a_g+...
5
votes
1answer
239 views

Zero-sum sets in union-closed families

The Davenport constant $D(G)$ of a finite abelian group $G$ is the minimum integer $n$ such that whenever $a_1, \ldots, a_n \in G$ (not necessarily distinct), there is a non-empty $I \subseteq [n]$ ...
6
votes
1answer
126 views

boundary of semihyperbolic groups

There are various definitions of boundary of a hyperbolic group. Which of those generalize to semi-hyperbolic groups (in the sense of Alonso and Bridson)? The example I have in mind is a semisimple ...
6
votes
1answer
154 views

Class number of Burnside groups

Let $B(m,n)$ be the Burnside group on $m$ generators of exponent $n$. Suppose the class number - the number of conjugacy classes - of $B(m,n)$ is finite. Does it imply that $B(m,n)$ is finite?
1
vote
1answer
101 views

Centralizer of a central involution in a simple group of Lie type

Let $G$ be a finite simple group of Lie type and $x$ be a central involution (that is, an involution which is contained in the center of a Sylow $2$-subgroup). Is it true that, if $y$ is another ...
1
vote
1answer
126 views

When the reduced $C^*$-algebra of $\Gamma$ admits character then $\Gamma$ is amenable [closed]

Suppose that $C^*_r(\Gamma)$ admits some character (homomorphism into $\mathbb{C}$)-here $\Gamma$ is discrete group and $C^*_r(\Gamma)$ is the closure of the image of the group ring $\mathbb{C}\Gamma$ ...
1
vote
0answers
91 views

to what extent is a reductive group hyperbolic?

The group $SL(2,F)$ where $F$ is a local nonArchemidian field is hyperbolic. Various generalizations of the notion of hyperbolicity have been studied in the literature (I've seen terms like "...
6
votes
3answers
328 views

Asymptotics for the number of abelian groups of order at most $x.$

The number of abelian groups of order $n$ (call it $a(n)$ is a studied subject (see http://oeis.org/A000688), but I can't seem to find any asymptotic results. Obviously, there is no asymptotic for $a(...
0
votes
0answers
58 views

HNN extension group with finitely generated base

Let $B$ be a group and let $A_1$ and $A_2$ subgroups of $B$ with $\phi :A_1\rightarrow A_2$ an isomorphism. Let $\left<t\right>$ be the infinite cyclic group, generated by a new element $t$. The ...
2
votes
1answer
317 views

One-dimension Algebraic groups

I am searching for a possible analogue of a result in algebraic groups in a non-commutative setting, so I am looking for different proofs of the following : Let $K$ be an algebraically closed field. ...
1
vote
1answer
134 views

p-groups with unique normal minimal subgroup

Have $p$-groups with a unique normal minimal subgroup been classified? Is there any article on the subject?