# Operation of GL_n(Z/bZ) [closed]

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

-

## closed as too localized by José Figueroa-O'Farrill, Ben Webster♦Apr 25 '10 at 12:51

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

First, this question is not really appropriate for MO. Second, the general linear group does not act on any given affine hyperplane. The subgroup which does is the affine linear group in one dimension lower. This acts transitively because of the translations, as in Robin's answer. – José Figueroa-O'Farrill Apr 25 '10 at 11:07
MO isn't intended for questions of the type that would be HW in an undergraduate class (whether they are or not), so your question has been closed. Some other sites that might work better for you are listed in the FAQ. – Ben Webster Apr 25 '10 at 12:54

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 $j>1$.