Classification of finite simple groups facts for kids

In mathematics, the classification of the finite simple groups is a huge achievement in group theory. It tells us that every finite simple group fits into one of a few main types. Think of simple groups as the basic building blocks for all other finite groups, much like prime numbers are the building blocks for all natural numbers.

This amazing proof took many years to complete. It involved thousands of pages of work by about 100 mathematicians!

What the Classification Theorem Says

The classification of the finite simple groups shows how all finite simple groups fit into different categories.

This classification helps mathematicians understand many things about finite groups. If you have a question about a finite group, you can often break it down into questions about these simpler building blocks. Then, you can check each type of simple group to find the answer.

A mathematician named Daniel Gorenstein announced the classification was done in 1983. But it turned out a small part of the proof was still missing. This missing part, about "quasithin groups," was finally completed in 2004 by Michael Aschbacher and Stephen Smith.

How the Proof Was Built

The proof of this theorem is very long and complex. It can be thought of as having several big parts.

Groups with Small "2-Rank"

Some simple groups are classified based on their "2-rank." This is a technical way to describe how many times you can divide the group's size by 2.

• Groups with 2-rank 0 are called "odd order groups." The Feit–Thompson theorem showed that these groups are always "solvable," meaning they are not simple (except for the smallest one).
• Groups with 2-rank 1 or 2 are also handled in special ways. For example, the Gorenstein–Walter theorem helped classify groups with a specific kind of "Sylow 2-subgroup" (a special subgroup related to the number 2).

These early parts of the proof often used "character theory," which is a way to study groups using numbers.

Groups of "Component Type"

Imagine a group has a special part inside it, called a "component." If this component is a smaller, known simple group, then the larger group is called "of component type." These are often groups of Lie type with odd characteristics, or alternating groups, plus some sporadic groups.

The idea here is like building with LEGOs. If you know what a smaller LEGO brick looks like, you can figure out what bigger structures it can be part of. Mathematicians used this idea to classify these groups by looking at their smaller simple parts.

Groups of "Characteristic 2 Type"

These groups are different. They are roughly like groups of Lie type over fields of "characteristic 2." This means they behave in a special way when you do math with them using the number 2.

These groups are also divided by their "rank."

• "Rank 1" groups are called "thin groups."
• "Rank 2" groups are called "quasithin groups." This was the last part of the proof to be completed in 2004.

For groups with higher ranks (3 or more), a special rule called the trichotomy theorem helps divide them into three types, making them easier to classify.

Making Sure Groups Exist and Are Unique

After classifying all the types of simple groups, mathematicians had to do two more things:

• Prove that each type of simple group actually exists.
• Prove that each type is unique (meaning there's only one group for each description).

This was a big task. For example, proving the existence and uniqueness of the monster group (the largest sporadic group) took about 200 pages! Sometimes, computers were used for these proofs, but later, simpler "by hand" proofs were found.

History of the Proof

The journey to classify finite simple groups began a long time ago.

• In 1832, Évariste Galois started thinking about "normal subgroups" and found some simple groups like the alternating groups.
• In 1861, Émile Mathieu discovered the first "sporadic groups," which were unexpected and didn't fit into the known families.
• In 1904, William Burnside proved that any non-abelian finite simple group must have a size divisible by at least three different prime numbers.
• A major breakthrough came in 1963 when Walter Feit and John Thompson proved the odd order theorem. This theorem showed that the only finite simple groups with an odd number of elements are the cyclic groups of prime order. This was a huge step, as it meant all other simple groups must have an even number of elements.
• In 1972, Daniel Gorenstein laid out a 16-step plan to finish the classification. This plan guided much of the work that followed.
• In 1982, Robert Griess built the monster group by hand, showing it truly existed.
• The final gap in the original proof was filled in 2004 by Michael Aschbacher and Stephen Smith, completing the classification of quasithin groups.

A Simpler Proof (Second Generation)

Because the first proof was so long and complicated, mathematicians have been working on a "second-generation classification proof." This new version aims to be simpler and shorter.

One reason a simpler proof is possible now is that we know the final list of all simple groups. The first group of mathematicians didn't know how many sporadic groups there were, so they had to use very general methods. Now, knowing the answer, they can use more direct and efficient techniques.

This new proof is still very long, but it's written in a more organized way. It also avoids repeating work that was done multiple times in the first proof.

Why This Classification Matters

The classification of finite simple groups is a very important result in mathematics. It has helped prove many other theorems, such as:

• The Schreier conjecture (about how groups can be built from smaller ones).
• The classification of 2-transitive permutation groups (groups that can move elements around in a very specific way).
• Frobenius's conjecture (about the number of solutions to certain equations in groups).

See also

O'Nan–Scott theorem