12
votes
1answer
249 views

Largest subset of $GL_n(p)$ in which pairwise subtraction is also in $GL_n(p)$

Suppose $X\subset \mathrm{GL}_n(p)$ is a set of invertible matrices such that for every $A,B\in X$ then also $A-B\in \mathrm{GL}_n(p)\cup \{0\}$. (If anyone knows a name for such sets I would be ...
1
vote
2answers
246 views

When is PSU(2,q^2) = PSL(2,q) ?

The context for this question is from page 284 - 287 of Berger's paper: ...
1
vote
0answers
237 views

Injective Mapping

Let y=Ax. A is a matrix n by m and m>n. Also, x gets its values from a finite alphabet. Elements of A and x can be complex numbers. How can i show if the mapping from x to y is injective for given A ...
2
votes
1answer
490 views

Linear algebra of finite abelian groups

If $f: V \to W$ is a surjective homomorphism of vector spaces, and we have fixed a basis for $V$, it is always possible to find a basis for $W$ such that the matrix associated to $\phi$ in the two ...
1
vote
1answer
282 views

trace of a matrix of finite order

Let $A$ be an $n$ by $n$ real matrix of order $d$. i.e. $d$ is the smallest positive integer greater than $1$ that makes $A^{d}=I_{n}$. The set of trace zero real matrices form $n^{2}-1$ dimensional ...
4
votes
1answer
274 views

Aschbacher classes and $\mathbb{F}_p$-subspace stabilizers in classical linear groups

I am reading the Kleidman-Liebeck book ("The subgroup structure of the finite classical groups") which is about the Aschbacher classification of maximal subgroups of the classical almost simple ...
2
votes
1answer
197 views

Nonsingularity of certain block matrices

I am completely stuck in the following linear algebra problem. Consider a finite group $H$ and $N\times N$-matrices $M_{g,h}$ with entries in $\mathbb{Z}$ for all $g,h\in H$. Assume $\sum_{h\in H} ...
4
votes
1answer
1k views

Finite subgroups of GL_n(C)

A finite groups of $\mathrm{GL}_n(\mathbb C)$ of exponent $m$ necessarily have order $C$ verifying $C\leqslant m^n$ and $n! m^n$ divides $C$, but this condition is not sufficient, for instance ...
6
votes
2answers
653 views

Linear algebra and regular orbits

If $A$ is an $n\times n$ matrix over a field, and $A^{k} = I$, with $k$ the least positive integer such that this occurs, then must there be some vector $v$ such that $\{v,Av,A^{2}v,\dots,A^{k-1}v\}$ ...