Example of an nonidentity element in the kernel of the map.

Question

This question is related to my previous question here.

Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: \text{SL}_n(\mathbb{Z}) \to \text{SL}_n(\mathbb{Z}/m\mathbb{Z})$.

What is an example of an element $g \in \text{Ker}(F)$, $g \neq \text{Id}$?

What does this map do, replace each item $a_{ij}$ in the matrix with $a_{ij} \pmod m$? — pjs36, Aug 25 '15 at 16:42

score 0 · Answer 1 · edited Aug 25 '15 at 19:52

0

The kernel just consists of matrices congruent to the identity mod $m$, so an example for $n=2$ would be: $\pmatrix{1+m & 2m\\ -3m & 1-5m}$.

edited Aug 25 '15 at 19:52

user26857

53,190

answered Aug 25 '15 at 18:16

oxeimon

12,569

Example of an nonidentity element in the kernel of the map.

1 Answers1