Classification of finite simple groups

The Classification of finite simple groups is a mega-theorem which states that every finite simple group belongs to one of eighteen infinite families of simple groups, or to one of 26 sporadic simple groups.

The eighteen families

Here are the families, up to isomorphism. Note that these families are one-parameter, two-parameter or three-parameter families. Each parameter varies either over prime numbers or over natural numbers. Many of the families have a few small exceptions that turn out not to be simple groups.

Also, some of these families have intersections, i.e., there are some groups that occur in multiple families. The intersection of any two families is finite: there are only finitely many groups that are simultaneously in two distinct families.

1. The cyclic groups of prime order (one parameter: prime number): These are the only simple Abelian groups. The set of these groups is in one-to-one correspondence with the set of prime numbers. Since there are infinitely many primes, there are infinitely many such groups. The order of the group equals the prime parameter. For full proof, refer: No proper nontrivial subgroup implies cyclic of prime order, prime order implies no proper nontrivial subgroup
2. The alternating groups of degree at least $5$ (one parameter: natural number): The alternating group of degree $n$, denoted $A_n$, is defined as the subgroup of the symmetric group on $n$ letters comprising the even permutations. The proof of their simplicity is inductive, using as base case the fact that $A_5$ is simple. The order of the group equals $n!/2$. For full proof, refer: A5 is simple, alternating groups are simple
3. The projective special linear group of a given order over a finite field (three parameters: natural number giving the order of matrices, prime number giving the characteristic, and natural number giving the exponent to which the prime needs to be raised to give the order of the field): The group with parameters $n,p,r$ is defined as $PSL(n,p^r)$ or $PS{L}_n(p^r)$. This is simple except when $n=2$ and $k$ has two or three elements. The order is equal to the product $\frac{p^{rn(n-1)/2}\prod _{i=2}^n(p^{ir}-1)}{\mathrm{g}\mathrm{c}\mathrm{d}(n,p^r-1)}$.For full proof, refer: Projective special linear group is simple
4. The projective special orthogonal group of a given order over a finite field (three parameters: natural number giving the order of matrices, prime number giving the characteristic, and natural number giving the exponent to which the prime needs to be raised to give the order of the field): The group with parameters $n,p,r$ is defined as $PSO(n,p^r)$ or $PS{O}_n(p^r)$. This is simple except when $p=2$. For full proof, refer: Projective special orthogonal group is simple
5. The projective special unitary group of a given order over a finite field (three parameters: natural number giving the order of matrices, prime number giving the characteristic, and natural number giving the exponent to which the prime needs to be raised to give the order of the field): The group with parameters $n,p,r$ is defined as $PSU(n,p^r)$ or $PS{U}_n(p^r)$. For full proof, refer: Projective special unitary group is simple
6. The projective symplectic group of a given order over a finite field (three parameters: natural number giving the order of matrices, prime number giving the characteristic, and natural number giving the exponent to which the prime needs to be raised to give the order of the field): The group with parameters $n,p,r$ is defined as $PSp(n,p^r)$ or $PS{p}_n(p^r)$. For full proof, refer: Projective symplectic group is simple

The twenty-six sporadic simple groups

1. The five Mathieu groups.
2. The four Janko groups.
3. The three Conway groups.
4. The three Fischer groups.
5. The Higman-Sims group.
6. The McLaughlin group.
7. The Held group.
8. The Rudvalis group
9. The Suzuki sporadic group.
10. The O'Nan group.
11. The Harada-Norton group.
12. The Lyons group.
13. The Thompson group.
14. The Baby Monster group.
15. The monster group: This is the largest sporadic simple group.

References

Expository article references

• A brief history of the classification of the finite simple groups by Ronald Mark Solomon, Bulletin of the American Mathematical Society, ISSN 10889485 (electronic), ISSN 02730979 (print), Volume 38,Number 3, Page 315 - 352(Year 2001): An expository paper by Ronald Mark Solomon describing the 110-year history of the classification of finite simple groups.^{Weblink (PDF)More info}
• Paper:AschbacherCFSG^{More info}