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28 votes
1 answer
2k views

Integer matrices which are not a power

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}$In a group $G$, an element $g$ is said to be primitive if there is no $h \in G$ and integer $n >1$ such that $g = h^n$. (For clarification, I ...
Philippe Tranchida's user avatar
9 votes
1 answer
709 views

Automorphisms of a matrix in Smith normal form?

Added: Amritanshu Prasad's answer makes it clear that I am really asking for a description of the group of integer unimodular matrices $P$ such that $D^{-1}PD$ is also integer. These matrices are ...
Will Orrick's user avatar
  • 2,150
8 votes
1 answer
361 views

Invertible matrix with group ring coefficient

Before asking the question I do need some notations. $G$ a (torsion-free) group, $\mathbb{Z}^{´}=\mathbb{Z}[\frac{1}{2}]$ $R:= \mathbb{Z}[G]$, $R^{´}=\mathbb{Z}^{´}[G]$ group rings. $Mat_{n}(R)$ the ...
GSM's user avatar
  • 223
6 votes
0 answers
259 views

Diameter of finite rational matrix groups

Suppose $G$ is a finite subgroup of $\mathrm{GL}(n,\mathbb{Q})$. For a set $\mathcal{M} \subseteq G$ that generates $G$, define the $\mathcal{M}$-diameter $\mathit{diam}(G, \mathcal{M})$ of $G$ to be ...
Stefan Kiefer's user avatar
5 votes
3 answers
673 views

Finite index subgroup of $\mathbb{Z}^2$ that is invariant under a non-singular matrix

Let $M $ be a matrix in $ \operatorname{GL}(2, \mathbb{Z})$ that has at least one eigenvalue of absolute value strictly bigger than $1$. What are the finite index subgroups $H$ of $\mathbb{Z}^2$ such ...
ghc1997's user avatar
  • 823
3 votes
0 answers
132 views

the growth rate of poly-$\mathbb{Z}$ group

I am interested in the growth rate of the poly-$\mathbb{Z}$ group. Let $G$ be a poly-$\mathbb{Z}$ group, i.e $$G =(\dots((\mathbb{Z} \rtimes_{\phi_1} \mathbb{Z})\rtimes_{\phi_2} \mathbb{Z}) \rtimes_{\...
ghc1997's user avatar
  • 823
2 votes
1 answer
187 views

A closed subgroup $G$ of $\operatorname{GL}_2 \mathbb{Z}_\ell$ which surjects onto $\operatorname{GL}_2 \mathbb{F}_\ell$

Let $\ell \ge 5$ be a prime and $G$ be a closed subgroup of $\operatorname{GL}_2 \mathbb{Z}_\ell$ whose image in $\operatorname{GL}_2 \mathbb{F}_\ell$ is $\operatorname{GL}_2 \mathbb{F}_\ell$. Then $G ...
zom's user avatar
  • 185
2 votes
1 answer
512 views

Submatrices of matrices in $\mathrm{SL}(4, \mathbb{Z})$ with all eigenvalues equal to $1$ [closed]

This is a follow-up question to my question from Math Stackexchange (Thank you Dietrich Burde and Michael Burr for the help). Let $M\in \mathrm{SL}(4, \mathbb{Z})$ with all eigenvalues equal to $1$ (i....
ghc1997's user avatar
  • 823
1 vote
1 answer
252 views

Smith normal form and last invariant factor of certain matrices

I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts can provide some relevant insight. Suppose we have row vectors $x_1$, $x_2$ , $y_1$ , $y_2 ...
ghc1997's user avatar
  • 823
0 votes
1 answer
171 views

Prime index subgroups of $\langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle$ that is invariant under matrix $Q$

Let $Q $ be a matrix in $ \operatorname{GL}(2, \mathbb{Q}) $ and consider the group $G = \langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle := \langle Q^{i}(v) \mid i \in \mathbb Z, v \in \...
ghc1997's user avatar
  • 823