proof of simplicity of Mathieu groups
We give a uniform proof of the simplicity of the Mathieu groups , , and , and the alternating groups (for ), assuming the simplicity of and . (Essentially, we are assuming that the simplicity of the projective special linear groups is known.)
If , are any transitivity classes for the restricted action, let , , and such that . Then is a bijective map from onto (here we use normality). Hence any element of maps transitivity classes to transitivity classes. ∎
Hence it follows:
Let be a group acting primitively and faithfully on a set . Let be the stabilizer of some point , and assume that is simple. Then if is a nontrivial proper normal subgroup of , then is isomorphic to the semidirect product of by . can be identified with in such a way that is identified with , the action of becomes left multiplication, and the action of becomes conjugation.
Since is a normal subgroup of , it is either or .
If , then , and since is maximal and is proper, we have . Since is normal and stabilizes , then stabilizes every point (since the action is transitive). Since the action is faithful, , a contradiction. (This contradiction can also be reached by applying the corollary.)
Therefore, . So no element of , other than 1, fixes . Thus acts freely and transitively on . For any , if and , then , hence is in . Thus is generated by and . Since is normal and , is the (internal) semidirect product of by . ∎
Now we come to the main theorem from which we will deduce the simplicity results.
Let be a group acting faithfully on a set . Let and let be the stabilizer of . Assume is simple.
1. Assume the action of is doubly transitive, and let be a nontrivial proper normal subgroup of . Then is an elementary abelian -group for some prime . Furthermore, is isomorphic to a subgroup of , and is isomorphic to a subgroup of , the group of affine transformations of .
2. If the action of is triply transitive and , then any nontrivial proper normal subgroup of is an elementary abelian 2-group.
3. If the action of is quadruply transitive and , then is simple.
For part 1, use the identification of with given by the previous theorem. Since the action is doubly transitive, the action by conjugation of is transitive on . Therefore, all non-identity elements of have the same order, which must therefore be some prime . Hence is a -group. The center is nontrivial, and is preserved by all automorphisms. By double transitivity again, there is an automorphism taking any nontrivial element to any other; hence is abelian. Therefore is an elementary abelian -group.
For part 2, we know from part 1 that is isomorphic to an elementary abelian -group and acts as linear transformations of . Since the action of is triply transitive, the action of on the nonzero elements is doubly transitive. However, if , then the linearity of the action disallows double transitivity (if , then so we do not have complete freedom for any two elements since some element besides 0, and .)
(We note that when , we have the example , , .)
Here is an example illustrating part 2. The group acts triply transitively on , and the stabilizer of a point is , which is simple if . contains the normal subgroup of translations, an elementary abelian 2-group.
For part 3, note that the action of on , , is not triply transitive on nonzero elements, so the only conclusion left is that is simple. ∎
The Mathieu groups , , , and are simple.
We take it as known that is simple. Since has as point stabilizer, and has a triply transitive action on a set of elements, we may work our way inductively up to , using the previous theorem. The index (http://planetmath.org/Coset) of in is , which is not a power of 2. Hence in all cases, is simple. ∎
The alternating groups are simple for .
Since the natural action of on letters is quadruply transitive for , and the point stabilizer of is , we may apply the theorem to deduce the simplicity of the alternating groups , , from the simplicity of . ∎
|Title||proof of simplicity of Mathieu groups|
|Date of creation||2013-03-22 18:44:08|
|Last modified on||2013-03-22 18:44:08|
|Last modified by||monster (22721)|