# Maths - Categorising Groups

These pages show various groups categorised in various ways:

### Classification of Finite Simple Groups

We will see (on these pages) that there is a sort of meta-algebra of groups, what I mean by that is that is that we can 'multiply' or 'divide' whole groups (rather than just elements of those groups) to form other whole groups. This analogy is not exact in that the rules for this type of 'meta multiplication and division' is different from the type of operations that we are used to when applied to elements of groups and rings. However it does mean that, in the same way that we can 'factorise' integers and derive the prime numbers we can 'factorise' groups into subgroups. If we can't reduce a group into subgroups (apart from the trivial case of itself and the identity element) then it is known as a 'simple group'. In the same way that the structure of prime numbers is very complicated then the structure of simple groups is also very complicated.

A subgroup of this type is known as a quotient group or factor group (a special case is the normal subgroup)

Any simple group belongs to one of the following families of groups:

• A cyclic group with prime order.
• An alternating group of degree at least 5.
• Lie groups
• The classical Lie groups, namely the groups of projective special linear, unitary, symplectic, or orthogonal transformations over a finite field;
• The exceptional and twisted groups of Lie type.
• One of 26 sporadic simple groups

Where:

• 'order' means the number of elements in the group.

### History of Classification of Finite Simple Groups

As we have seen the structure of finite simple groups is very messy, the final sporadic group was not found until 1975. Then it was not until 1981 that the proof of the classification of finite simple groups was completed, that is the proof that the categories are complete and there are no more to be discovered. This is so complicated that even after 1981 gaps in the proof were still being found and filled in. Even now we can't be sure that there in not a hidden error.

This classification proof is perhaps the largest collaborative piece of pure mathematics ever attempted. It consists of over 10,000 pages, across about 500 journal articles, by over 100 different authors from around the world, it was without precedent.