Group for logicians

Definition
For logicians, in the theory of definability and stability for groups, a group is an abstract group along with possibly other functions and relations that may not be directly definable from the group operation. If there are no additional operations, logicians, call the group a pure group.

Importance
One of the reasons why logicians prefer to consider groups with (possibly) additional structure is that when we take a subgroup of a given group (even if it is pure) we do get some additional structure on the subgroup arising from its embedding in the bigger group. Thus, even if we start only with pure groups, we do end up with the more general notion of group with additional structure.