Questions tagged [nilpotent-groups]
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97 questions
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Does every finite solvable group contain a non-trivial Abelian subgroup, invariant under all semi-automorphisms?
A bijective map $f:G\to G$ of a group is called a semi-automorphism of $G$ if $f(xyx)=f(x)f(y)f(x)$ for all $x,y\in G$.
Question 1. Is it true that every finite solvable group contains a non-trivial ...
- 45.5k
3
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0
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Many degree $d$ nilpotent elements of quotients of polynomial rings and non-vanishing product
Generalization of this question.
Let $n$ be positive integer and $f_1(x_1,...,x_k), \dots, f_m(x_1,...x_k)$
polynomials with integer coefficients.
Let $K=\mathbb{Z}/n\mathbb{Z}[x_1,...,x_k]/\langle ...
- 25.8k
1
vote
0
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121
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Criterion for when certain closed 2-forms generates all n-forms
Let $\omega_1,\dots,\omega_r$ be closed 2-forms in $\Bbb R^m$.
I want to find a smallest number $n\ge1$ such that every $(n+1)$-form $\eta$ can be written as $\eta=\beta_1\wedge\omega_1+\dots+\beta_r\...
- 1,511
5
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1
answer
387
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Why do we care about residually solvable/nilpotent groups?
I'm interested in a certain property $X$. In the introduction to basically every paper on $X$ there's a paragraph that goes something like:
$X$ is related to residually-solvable groups in this way, ...
- 491
7
votes
1
answer
564
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Khovanskii's theorem in nilpotent groups?
In $\mathbb{Z}^d$, a classical result of Khovanskii states that for any finite set $A \subseteq \mathbb{Z}^d$, the sumset $hA := A + \cdots + A$ eventually agrees with a polynomial in $h$; that is, $|...
- 972
4
votes
1
answer
293
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Outer automorphism groups of nilpotent groups
I am interested in the class of finitely generated nilpotent groups. It seems that many things are known about the automorphism groups of such groups. However, I could not find much information on ...
- 141
8
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1
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336
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Finite index subgroup of nilpotent group formed by squares
Let $G$ be a finitely generated nilpotent group. Is there a finite index subgroup $H$ of $G$ such that $H\subseteq\{g^2;g\in G\}$?
Context: I considered this question while studying `actions' of the ...
- 13.4k
1
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0
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91
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Positivity of polynomial ergodic averages in nilpotent groups
Let $(X,\mu)$ be a probability space with an action $(T_g)_{g\in G}$ of a group $G$ by unitary transformations. Theorem 1.1 in Zorin-Kranich's paper implies that, if $G$ is nilpotent, $H$ is a (...
- 13.4k
0
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1
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151
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Normalizer of $\varphi$-invariant subgroup is $\varphi$-invariant
In a paper I'm writing, I'm proving the following two results:
Let $N$ be a finitely generated torsion-free nilpotent group and $H$ a subgroup of $N$. Suppose $\varphi$ is an automorphism of $N$ such ...
- 129
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79
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Homomorphism to a finite p-group/Lie ring Q: estimate on |Q|
Let $L$ be a finitely generated, torsion-free nilpotent group. Furthermore, assume it is also a Lie ring, i.e. a Lie algebra but over $\mathbb{Z}$. The correspondance between $L$ as a group and $L$ as ...
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11
votes
1
answer
480
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A question on groups having a subgroup which fixes a vector in every irreducible representations
Given a finite group $G$, I am interested in finding a non-trivial proper subgroup $H$ of $G$ such that $\mathrm{Ind}_H^G\mathbf{1}$ contains all the irreducible representations of $G$, that is, ...
- 419
3
votes
0
answers
139
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Lattice in a simply connected nilpotent Lie group
Given a connected and simply connected nilpotent Lie group $N$ with a left invariant metric, we assume that there is a lattice $\Gamma$ of $G$. Let $B_1(e)$ be the $1$-ball at the identity element in $...
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3
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1
answer
546
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Is Malcev completion an embedding?
The Malcev completion of an abstract group $G$ over a field $k$ of characteristic zero is defined by
$$\hat G = \mathbb{G}(\widehat{k[G]}) ,$$
the group-like part of the completed (by the augmentation ...
- 967
9
votes
3
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Finding a nilpotent, infinite, f.g., virtually abelian, irreducible integer matrix group
I was wondering if anyone here knows of an example of a group $M \leq \mathrm{GL}_n(\mathbb{Z})$ which is
nilpotent,
infinite,
finitely generated,
virtually abelian,
irreducible (over $\mathbb{Z}$ or ...
- 5,792
6
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0
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Equation in a nilpotent group
Let $G$ be a nilpotent group of class at most $r$
(that is, $\gamma^{r+1}G=1$).
Let elements $g_1,\dotsc,g_n\in G$ be fixed.
We are interested in the set $V\subseteq\mathbb Z^n$ of solutions $x=(x_1,\...
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