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let $K$ be a field, $n \geq 1$. denote $E_{i,j}$ the elementary matrix having $1$ on the diagonale and in the entry $(i,j)$, and $E_i(a)$ the elementary matrix $diag(1,...,a,...,1)$. you know that $GL_n(K)$ is generated by these matrices, but what relations do we need in order to get a presentation for $GL_n(K)$?

here are some relations, which correspond to simple relations about row operations:

• $E_i(1)=1$
• $E_i(ab) = E_i(a) E_i(b)$
• $E_i(a) E_j(b) = E_j(b) E_i(a)$
• $(E_j(-1) E_{ij})^2=1$
• $E_j(a+b)^{-1} E_{ij} E_j(a+b) = E_j(a)^{-1} E_{ij} E_j(b) E_i(a)^{-1} E_{ij} E_j(a)$
• $(E_{ji} E_{ij} E_{ji} E_j(-1))^2=1$

are these all relations? how can we prove that?

EDIT: Mariano has given a counterexample when $K = \mathbb{F}_2$. well, how can we fix this? add more relations? incorporate the structure of $K$ as a ring? what about concrete examples such as $K=\mathbb{Q}$?

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# presentation for GL(n,K)

let $K$ be a field, $n \geq 1$. denote $E_{i,j}$ the elementary matrix having $1$ on the diagonale and in the entry $(i,j)$, and $E_i(a)$ the elementary matrix $diag(1,...,a,...,1)$. you know that $GL_n(K)$ is generated by these matrices, but what relations do we need in order to get a presentation for $GL_n(K)$?

here are some relations, which correspond to simple relations about row operations:

• $E_i(1)=1$
• $E_i(ab) = E_i(a) E_i(b)$
• $E_i(a) E_j(b) = E_j(b) E_i(a)$
• $(E_j(-1) E_{ij})^2=1$
• $E_j(a+b)^{-1} E_{ij} E_j(a+b) = E_j(a)^{-1} E_{ij} E_j(b) E_i(a)^{-1} E_{ij} E_j(a)$
• $(E_{ji} E_{ij} E_{ji} E_j(-1))^2=1$

are these all relations? how can we prove that?