TITLE: In group theory, what is meant by the structure of a group?
QUESTION [6 upvotes]: In group theory, what is meant by the structure of a group?
I have an intuitive idea of this (well, I think). e.g.: the structure relates to the operation of the group and its set. 
But is there a formal definition of the idea of structure? or it's really just a qualitative term to describe a group?  (and, in fact, many other mathematical objects, like Vector spaces, etc...)

REPLY [2 votes]: There exists a formal definition of a notion of structure defined on a set. You can read it in Bourbaki theorie des ensembles, chapitre 4 or here,
https://en.wikipedia.org/wiki/Mathematical_structure