Classification of finite simple groups: Difference between revisions

Revision as of 15:53, 1 January 2009

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.

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. For full proof, refer: No proper nontrivial subgroup implies cyclic of prime order, prime order implies no proper nontrivial subgroup
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. For full proof, refer: A5 is simple, alternating groups are simple
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 $PSL(n,p^{r})$ .

The twenty-six sporadic simple groups

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 ==The eighteen families==
-Here are the families, up to isomorphism:
+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.
-# The [[group of prime order|cyclic groups of prime order]]: 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. {{proofat|[[No proper nontrivial subgroup implies cyclic of prime order]], [[prime order implies no proper nontrivial subgroup]]}}
-# The [[alternating group]]s of degree at least <math>5</math>. The alternating group of degree <math>n</math>, denoted <math>A_n</math>, is defined as the subgroup of the [[symmetric group]] on <math>n</math> letters comprising the [[even permutation]]s. The proof of their simplicity is inductive, using as base case the fact that <math>A_5</math> is simple. {{proofat|[[A5 is simple]], [[alternating groups are simple]]}}
+# The [[group of prime order|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. {{proofat|[[No proper nontrivial subgroup implies cyclic of prime order]], [[prime order implies no proper nontrivial subgroup]]}}
+# The [[alternating group]]s of degree at least <math>5</math> (one parameter: natural number): The alternating group of degree <math>n</math>, denoted <math>A_n</math>, is defined as the subgroup of the [[symmetric group]] on <math>n</math> letters comprising the [[even permutation]]s. The proof of their simplicity is inductive, using as base case the fact that <math>A_5</math> is simple. {{proofat|[[A5 is simple]], [[alternating groups are simple]]}}
+# 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 <math>n,p,r</math> is defined as <math>PSL(n,p^r)</math>.
 ==The twenty-six sporadic simple groups==
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