# Thread: order preserving group homomorphism

1. ## order preserving group homomorphism

say you have two subgroups $G\text{ and }H$ and an order preserving group homomorphism $\eta:G\rightarrow H$ which maps the identity of $G$ to the identity of $H$

Can we show that $\eta(g)=g$ for all $g\in G$ so that we can identify $G$ as a subgroup of $H$ and $\eta$ is then just the inclusion?

2. Originally Posted by Mauritzvdworm
say you have two subgroups $G\text{ and }H$ and an order preserving group homomorphism $\eta:G\rightarrow H$ which maps the identity of $G$ to the identity of $H$

Can we show that $\eta(g)=g$ for all $g\in G$ so that we can identify $G$ as a subgroup of $H$ and $\eta$ is then just the inclusion?

The condition $\eta(g)=g\,,\,\forall g\in G$ is very strong, and it'd mean $G\subset H$ (i.e. not only as identification but as actual subset of a set...).

But from the given data yes: $\eta(g)=1\Longleftrightarrow g=1\Longleftrightarrow \eta\,\,is\,\,1-1$ and thus G is embedded in H.

Tonio