I'm fairly sure I'm not going to pose this question using the appropriate terminology so I will do my best to describe.
I'm interested in studying isomorphisms between two Groups (Albelian if possible) using the function $i:G\rightarrow H$ that preserve the distance between two two pairs of objects in the respective sets.
So let the two groups be $\left(G,.\right)$ and $\left(H, \times\right)$ and let the distance function between points in $G$ and $H$ map to an ordered Field $F$. For convienience let's say that $F = \mathbb{R}$. So let $d_{G}\left(x,y\right)$ and $d_{H}\left(x,y\right)$ be the distance functions for $G$ and $H$ respectively. Then what I would like to study is group isomorphisms that preserve distance between objects in their respective sets, i.e.
$\forall g_{1}, g_{2} \in G$
$d_{G}\left(g_{1}, g_{2}\right) = d_{H}\left(i\left(g_{1}\right), i\left(g_{2}\right)\right)$
is this studied in Abstract Algebra or elsewhere?
The motivation for my question comes from a big part of my PHD work in Natural Language Processing/Computational Linguistics. A common approach to assess word similarity is to take words (say in English) and map them onto Mathematical objects (High Dimensional Real Vector Spaces + Neural Networks are most commonly used) and then assess similarity between words using distance measures defined for the Mathematical Objects.
In order for this mapping to work perfectly (or as close to) we must ensure that the distances between respective pairs of words and say their corresponding vectors is preserved.
Any info on whether this has been studied and if so where to look would be greatly appreciated.
Thanks