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Results tagged with linear-algebra
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user 35416
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
14
votes
Algebra with a certain abelian group as the multiplicative group
I am going to assume that by "algebra" you simply mean a ring.
The answer is "no", in general. For example $\mathbb{Z}/5\mathbb{Z}$ is not the unit group of a ring. Indeed, suppose it was the unit gro …
- 13k
13
votes
Conjugacy classes of PGL(3,Z)
I will work in ${\rm GL}$ instead of ${\rm PGL}$.
The corresponding question over ${\rm GL}_3(\mathbb{Z})$ is essentially$^1$ equivalent to asking how many faithful $\mathbb{Z}[G]$-modules, free of r …
- 13k
5
votes
Heisenberg group over the Gaussian integers
A quick google search produced this paper. It gives generators and relations for the Heisenberg group over rings of integers of quadratic fields and discusses its representations.
- 22.8k
5
votes
Structure theorem for finitely generated Z[G] modules
The answer to your question is 'no'. Even if you limit yourself to modules that are free over $\mathbb{Z}$, there is no classification known. Indeed, if your abelian group is not cyclic or its order i …
- 13k
11
votes
Accepted
Torsion in GL_n(Z)
Here is a loose collection of partial answers, most of which address a simpler question than that Andy actually asked. I am particularly focusing on Andy's question in the comments "can we classify al …
- 13k
7
votes
Proof that bases etc. exist in early linear algebra course?
Dear Matthew, I have a couple of different remarks about the question and about the assumptions you seem to be making.
You write "I'm tempted to skip on to linear maps, matricies etc. which seems mo …
- 13k