An algebraic structure is something with a set, operation, and identities or axioms. They help us understand what different mathematical objects have in common and allow us to understand things more abstractly. Essentially, it’s just another level of generalisation beyond simple variables.
Note on notation
Often a generic operation is given as , but can also be written as , , as well as essentially any other symbol. Thus it should be clearly defined to avoid confusion.
Groups
Definition-of-a-Group
If a set withholds these four axioms, then that set is the type of algebraic structure called a group. Some of these groups follow an additional axiom which makes it an abelian group (commutativity, i.e. the order of operations does not affect the result). Often it is easy to disprove that a set is a group with a counter-example, but proving a set is a group is involves a rigorous proof of each axiom.
An informal example
The integers . If we take two integers and in , is also in ( is closed). If you add three integers together, whether you initially sum the first two or the last two, it doesn’t affect the result ( is associative). Adding to any integer doesn’t affect the result ( has the identity element ). For each integer, there is another integer which brings the first integer back to zero, i.e. for every in , there is (every element in has an inverse).
Note that groups aren’t all this clearly algebraic, there are also a lot of geometric groups, such as the transformations of a square .