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.
Best Answer
I am not really familiar with the 'algebraic system' terminology, but it seems as though the difference between that and an 'algebraic structure' is that an algebraic system can be a set with any binary operation on that set, whereas the operation on a set in an algebraic structure must conform to certain rules (in the case of groups, inverses, associativity, and identity).
Maybe the easiest example is the integers. As a set, $\mathbb{Z}$ is just a collection of elements which have nothing to do with each other - $0$ is no more special than $79$, $10$ and $-10$ have no particular relationship. But once we consider the integers as a group under the binary operation 'addition' (which I think we're all familiar with) we are able to see that $10+(-10)=0$ and $z + 0 = z$ for every $z\in \mathbb{Z}$.