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

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9
votes
0answers
188 views

Cohomology of central extensions of groups

Let G be a central extension of a finite group H by $Z/2$. I need an explicit description of the differentials $d_2$ and $d_3$ in the Lyndon-Hochschild-Serre spectral sequence which converges to the ...
4
votes
0answers
95 views

How to enumerate a discrete group of matrices by their Frobenius norm?

Suppose I have a discrete group $G<\mathrm{SL}_2(\mathbb{C})$, and it is finitely generated by some known generators. That is, $G=\langle g_1,\dots,g_n\rangle$. The Frobenius norm of a matrix ...
1
vote
1answer
189 views

What are the finite subgroups of $\operatorname{Sp}_{2n}(\mathbb{Z})$?

I've read the following question: Finite subgroups of ${\rm SL}_2(\mathbb{Z})$ (reference request) and it made me wonder. It's easy to see that ...
3
votes
1answer
156 views

Braid group: Can a left-twist increase the number of right twists?

Disclaimer: This question was first posted on math.se without any answer. This is something that naturally occurs in my research, but I am no expert on this - it feels like a natural question so I am ...
0
votes
1answer
121 views

The relationship between $p$-solvable Group and solvable group [closed]

Can anyone please tell me The relationship between $p$-solvable Group and solvable group.and find an example of a $p$-solvable group that is not solvable group or vice-versa.
7
votes
0answers
218 views

A “direct” proof that hyperbolic groups are not amenable

I am looking for a proof that a finitely generated hyperbolic groups is non-amenable [unless it is virtually cyclic] which is as "metric/combinatoric" as possible. Here are the two proofs I am aware ...
4
votes
1answer
117 views

A decomposition of $w_0$ which is similar to the reduced decomposition

Some basic definitions about reduced decomposition: In the symmetric group $S_n$, let $s_i$ denote the adjacent transposition $(i,i+1),i\in \{1,2,\cdots,n-1\}.$ Since $S_n$ is generated by adjacent ...
5
votes
1answer
167 views

A question on the commutativity degree of the monoid of subsets of a finite group

The commutativity degree $d(G)$ of a finite group $G$ is defined as the ratio $$\frac{|\{(x,y)\in G^2 | xy=yx\}|}{|G|^2}.$$It is well known that $d(G)\leq5/8$ for any finite non-abelian group $G$. If ...
1
vote
2answers
105 views

Any analysis on phase of eigenvalue of unitary matrix?

I understand that there are invariant Haar measure for eigenvalues of unitary matrix. I further understand that absolute value of eigenvalues of unitary matrix is 1. But, I could not find any analysis ...
0
votes
2answers
509 views

Is the absolute Galois group the same as the automorphism group? [closed]

Is the absolute Galois group $\mathrm{Gal}(\overline{\mathbf{Q}}|\mathbf{Q})$ the same as the group $\mathrm{Aut}_{\mathbf{Q}}(\overline{\mathbf{Q}})$ the automorphism group in the category of ...
2
votes
2answers
436 views

For a ring R, does $GL_n(R)$ embed into $GL_m(F)$ for some field F?

Suppose $R$ is a noncommutative ring. What is the sufficient and necessary condition for $GL_n(R)$ embedding into $GL_m(F)$ for some field $F$? In particular, what if $R$ is the group ring ...
0
votes
0answers
77 views

Embedding a cancellative monoid into another in such a way that $|X-x|=|X|$, where $X$ is a fixed finite set and $x\in X$

Preliminaries. Let $\mathbb A = (A, +)$ be a possibly non-commutative semigroup. For $X, Y \subseteq A$ we set $$ X - Y := \{a \in A: a + y \in X\text{ for some }y \in Y\}, $$ which is just the usual ...
3
votes
1answer
98 views

free subgroups of the fundamental group of nonorientable surfaces

Sorry for this possibly trivial or stupid question, but I'm very far from being an expert in Algebra. Let G be the fundamental group of the nonorientable surface of even rank n=2k (n generators, 2n ...
6
votes
1answer
195 views

Embedding a cancellative monoid into another in such a way that a prescribed element becomes left-invertible

Let $\mathbb A = (A, +_A)$ be a cancellative, but possibly non-commutative, monoid with identity $0$, and fix an element $x \in A$. Does there always exist a cancellative monoid $\mathbb B = (B, +)$ ...
7
votes
1answer
243 views

Does the Fano plane “embed” in the complex projective plane?

$PSL(2,7)$ acts on the projective plane over $\mathbb{F}_2$ (the Fano plane) through its identification with $GL(3,2)$. It also acts on the projective plane over $\mathbb{C}$ through either of its ...
1
vote
0answers
123 views

Which conjectures are proved for sofic groups? [closed]

Which conjectures about groups are resolved in case of sofic groups? I know two examples: Kaplansky's direct finiteness conjecture (proved by Gabor Elek). Some versions of Ornstein's isomorphism ...
31
votes
1answer
559 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
75 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
176 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
207 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
65 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
701 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
104 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
114 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: ...
5
votes
0answers
101 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
65 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
64 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 ...
1
vote
0answers
173 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
104 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
245 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
304 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
191 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 ...
2
votes
0answers
71 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
215 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
259 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
446 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
184 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 ...
5
votes
1answer
182 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
530 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
106 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 $$ ...
2
votes
0answers
86 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 ...
0
votes
1answer
292 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
655 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
83 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 ...