General linear group:GL(2,Z4)

Definition
This group can be defined in the following equivalent ways:


 * 1) It is the group $$GL(2,\mathbb{Z}_4)$$ or $$GL(2,\mathbb{Z}/4\mathbb{Z})$$, i.e., the defining ingredient::general linear group of degree two over the ring of integers modulo $$4$$.
 * 2) It is the group $$GL(2,\mathbb{F}_2[t]/(t^2))$$, i.e., the general linear group of degree two over the ring $$\mathbb{F}_2[t]/(t^2)$$.

Note that although the rings in question are different, the corresponding general linear groups are isomorphic. This behavior is specific to the cases $$p = 2$$ and $$p = 3$$. The isomorphism can be traced to the fact that the quotient map:

$$GL(2,\mathbb{Z}/4\mathbb{Z}) \to GL(2,\mathbb{Z}/2\mathbb{Z})$$

has a section, i.e., the corresponding short exact sequence splits. See also isomorphic general linear groups not implies isomorphic rings.