simplicity of the alternating groups
If , then the alternating group on symbols, , is simple.
It is easy to check that for every permutation we have
Two preliminary results will also be necessary.
The set of cycles of length generates .
A product of -cycles is an even permutation, so the subgroup generated by all -cycles is therefore contained in . For the reverse inclusion, by definition every even permutation is the product of even number of transpositions. Thus, it suffices to show that the product of two transpositions can be written as a product of -cycles. There are two possibilities. Either the two transpositions move an element in common, say and , or the two transpositions are disjoint, say and . In the former case,
and in the latter,
This establishes the first lemma. ∎
If a normal subgroup contains a -cycle, then .
In case is odd, then (because ) we can choose some transposition disjoint from so that
where is even. This means that contains all -cycles, as . Hence, by previous lemma as required. ∎
Proof of theorem.
Let be a non-trivial normal subgroup. We will show that . The proof now proceeds by cases. In each case, the normality of will allow us to reduce the proof to Lemma 2 or to one of the previous cases.
Suppose that there exists a that, when written as disjoint cycles, has a cycle of length at least , say
Upon conjugation by , we obtain
Hence, , and hence also. Notice that the rest of the cycles cancel. By Lemma 3, .
Suppose that there exists a whose disjoint cycle decomposition has at least two cycles of length 3, say
Conjugation by implies that also contains
Hence, also contains . This reduces the proof to Case 1.
Suppose there exists a of the form . Conjugating by with distinct from (at least one such exists, as ) yields
Hence . Again, Lemma 3 applies.
Suppose that contains a permutation of the form
This time we conjugate by .
which reduces the proof to Case 2.
Since there exists at least one non-identity , and since this is covered by one of the above cases, we conclude that , as was to be shown.
|Title||simplicity of the alternating groups|
|Date of creation||2013-03-22 13:07:57|
|Last modified on||2013-03-22 13:07:57|
|Last modified by||rmilson (146)|