Question

I want to show, that $GL_n(\mathbb{Z}/b\mathbb{Z})$ operates transitively on $X = \{ (v_1, \ldots, v_n) \in (\mathbb{Z}/b\mathbb{Z})^n \ | \ v_1\mathbb{Z}/b\mathbb{Z} + \ldots + v_n\mathbb{Z}/b\mathbb{Z} = \mathbb{Z}/b\mathbb{Z}\} $
($b$ is an integer.)

IOW

1. $A \in GL_n(\mathbb{Z}/b\mathbb{Z}), x \in X \Rightarrow Ax \in X$
2. For all $x,y \in X$ exists $A \in GL_n(\mathbb{Z}/b\mathbb{Z})$ such that $x = Ay$.

I have no idea, where to start.

Thanks
-elfinit

Answer 1

Any transformation $$(v_1,\ldots,v_n)\mapsto (v_1,\ldots,v_{j-1},v_j+av_k,v_{j+1},\ldots,v_n)$$ for $j\ne k$ is achievable by means of some such matrix. It suffices to reduce an admissible vector to $(1,0,\ldots,0)$ by means of a sequence of such reductions. I would do it in three stages

1. Make $v_n$ into a unit in $\mathbb{Z}/b\mathbb{Z}$;

2. Make $v_1=1$;

3. Make all $v_j=0$ for $j>1$.