The group name in taxonomy for a single kind of living thing that can reproduce and whose offspring can also reproduce is a species.

Here's why:

* Species is the fundamental unit of classification in biology.

* It defines a group of organisms that can interbreed and produce fertile offspring. This means that their offspring are also capable of reproducing.

* Members of the same species share a common gene pool and can exchange genetic information.

Let me know if you'd like to know more about the different levels of classification in taxonomy!