Simple groups are the building blocks of finite groups. After decades of hard work, mathematicians have finally classified all finite simple groups. Today we talk about why simple groups are so important, and then cover the four main classes of simple groups: cyclic groups of prime order, alternating groups An (n bigger than 4), groups of Lie type, and the 26 sporadic groups, including the Monster Group.
Composite numbers are the product of two or more prime numbers. Prime numbers cannot be factored, or “broken down” into smaller whole numbers. Similarly, in chemistry, every molecule is a collection of atoms bound together by chemical bonds. Atoms, like prime numbers, are considered indivisible—the simplest unit of matter. Identifying the building blocks of something and then how you can combine them can be a useful way to catalog the essential elements of many fields of study.
This reasoning can be extended to group theory. Simple groups can be thought of as analogous to prime numbers, or atoms. Simple groups are the building blocks of finite groups. They are those groups that have no proper, normal subgroups. Remember, if a group has a normal subgroup, then you can create a factor group. Since simple groups do not have any normal subgroups, you cannot create a factor group from them.
In this video, we discuss:
▪ the various types of simple groups
▪ why they are organized in this way
▪ how to distinguish them from each other (look out for the monster group)