Throughout this post, $latex k$ is a field and $latex k^{\times}:=k \setminus \{0\},$ the multiplicative group of $latex k.$ We have already seen the general linear group $latex \text{GL}(n,k)$ and the special linear group $latex \text{SL}(n,k)$ in this blog several times. The general linear group $latex \text{GL}(n,k)$ is the (multiplicative) group of all $latex n…

Abstract Algebra Blog

31.03.2024

A matrix is nilpotent iff the trace of every power of the matrix is zero | Abstract Algebra

If $latex A$ is a square matrix with entries from a field of characteristic zero such that the trace of $latex A^m$ is zero for all positive integers $latex m,$ then $latex A$ is nilpotent. This is a very well-known result in linear algebra and there are at least two well-known proofs of that (see…

Abstract Algebra Blog

13.04.2024

The Cartan-Brauer-Hua theorem | Abstract Algebra

For a division ring $latex D,$ we denote by $latex Z(D)$ and $latex D^{\times}$ the center and the multiplicative group of $latex D,$ respectively. Let $latex D_1$ be a division ring, and suppose that $latex D_2 $ is a proper subdivision ring of $latex D_1,$ i.e. $latex D_2$ is a subring of $latex D_1, \…

Abstract Algebra Blog

16.04.2024

If |G/Z(G)| = n, then G is n-abelian | Abstract Algebra

As we defined here, given an integer $latex n,$ we say that a group $latex G$ is $latex n$-abelian if $latex (xy)^n=x^ny^n$ for all $latex x,y \in G.$ Here we showed that if $latex G/Z(G),$ where $latex Z(G)$ is the center of $latex G,$ is abelian and if $latex |Z(G)|=n$ is odd, then $latex G$…

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Group homomorphisms from GL(n,k) to the multiplicative group of k | Abstract Algebra

A matrix is nilpotent iff the trace of every power of the matrix is zero | Abstract Algebra

The Cartan-Brauer-Hua theorem | Abstract Algebra

If |G/Z(G)| = n, then G is n-abelian | Abstract Algebra

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