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The Beautiful Abstract Structures Known as Finite Groups
Finite group theory is both beautiful and elegant — here’s a simple and pretty introduction
7 min read3 days ago
Groups are one of the most fundamental algebraic structures that mathematicians work with. They are defined to reflect structures and rules that we find all around us when we are trying to solve mathematical problems. Let’s start with a reminder of the definition of a group, and an illustrative example.
Definition: A group is a set of objects G and an operation on those objects * for which the following rules are obeyed:
- Rule of closure. If g, h are in G then g*h is in G
- Rule of associativity. For g, h and k in G, (g*h)*k= g*(h*k)
- Rule of identity. There exists an element e in G such that e*g = g*e = g for all elements g in G. That is, there is an element of G that, when operated on any element of G, does not change that element.
- Rule of inverses. For every element g of G, there exists a unique inverse element g` in G for which g*g`=g`*g=e.
Examples: The integers under the operation of addition is a group. It’s identity is zero, and the inverse of any element is its negation. The set of all rational numbers (excluding zero) is a group under multiplication. It’s identity is 1 and the inverse of any element a/b is its reciprocal b/a.
