I am learning about Groups from a discrete mathematics textbook for a computer science course by Grimaldi.
An algebraic system is defined as a set along with operations on elements of the set. An algebraic structure is defined as a set, operations on elements of the set and relations between elements of the set which leads to a structure on the elements of a set.
The author then goes on to talk about groups as being algebraic structures(so does wikipedia). I am failing to see what structure a group imposes on a set it is defined on. This is making me think about a group an algebraic system instead of an algebraic structure.
What should I see in a group that will help me relate it the definition of an algebraic structure? What should I understand by "structure on the elements of a set"? Explanation with examples would help a lot.
EDIT - Exact definitions as given in the textbook
An algebraic system is a system consisting of a nonempty set $A$ and one or more n-ary operations on the set $A$. It is denoted by $ \langle A, f_1, f_2, ... \rangle$.
An algebraic structure is an algebraic system, $\langle A, f_1, f_2, ... , R_1, R_2, ...\rangle$, wherein addition to operations $f_i$, the relations $R_i$ are defined on A. This leads to a structure on the elements of A.