Connected group for logicians

Generic notion
A group (possibly with additional structure and relations) is said to be connected or irreducible if it has no proper definable subgroup of finite index. Here by definable subgroup we mean subgroup definable with respect to the first-order theory of the group.

Metaproperties
The more structure we impose on a group, the harder it is for the group to remain connected. In other words, if a group is connected with a certain amount of additional structure, then it will continue to remain connected when that additional structure is removed.