A *ring* is an algebraic structure modelled after the integers and their arithmetic properties. Formally, a ring is defined as a structure consisting of a nonempty set, **S**, and two operations; addition(**+**) and multiplication(*****) satisfying the following properties for all *x*, *y*, *z* in **S**:

*x*+*y*is also a member of**S**(*closure under addition*)*x*+*y*=*y*+*x*(additive*commutativity*)*x*+ (*y*+*z*)= (*x*+*y*) +*z*(*associativity with respect to addition*)- There is an element
**0**, such that*x*+**0**=**0**+*x*=*x*for all*x*…

Pierre de Fermat, a seventeenth century French lawyer and mathematician needs no introduction, he is famous for the notoriously hard conjecture, Fermat’s Last Theorem(FLT), which took over three centuries and a half to prove after a series of failed attempts by cranks and professional mathematicians alike.

“*It is impossible to separate a cube into two cubes, a fourth power into two fourth powers, or generally, any power above the second into two powers of the same degree*”, Fermat wrote in the margin of his copy of an ancient math text. What that means in mathematical notation is that there are…

*A group is a monoid in which every element has an inverse.*

Fact of life.

I always chuckle when I think of that college textbook definition of a group. Innocently waggish though very correct.

While that opening definition of a group is correct, it doesn’t immediately convey the essence of a group, especially to a math noob.

Groups are ubiquitous objects in mathematics with diverse practical and theoretical applications. We are going to be looking at the meaning of a group and how the idea develops, but first, a look at sets and binary operations.

One of the fundamental notions…

Programmer. Mathematician. Thinker. Researcher at ZetaLabs, Lagos, Nigeria.