examples of infinite simple groups
Let be a set and let be a function. Define
For permutation , the set will play the role of a ,,bridge” between the infinite world and the finite world.
Let denote the group of all permutations on (with composition as a multiplication). For , subset will be called -finite iff is finite and . This is equivalent to the fact, that is finite and if , then .
It is easy to see, that if and is -finite, then . Thus, we have well defined permutation (on a finite set) by the formula .
Lemma. For any subset and any such that is -finite and -finite we have that is -finite and
holds, because is well definied (since is -finite) and the operation does not change the formulas of functions.
Now we can talk about the sign of a permutation. For define
It can be easily checked, that is well defined (indeed, sign depends only on those for which ). Furthermore, it follows directly from the definition, that
is a group homomorphism (in we have standard multiplication). Define
Proof. Assume, that is not simple and let be a proper, nontrivial, normal subgroup. For a subset define
Obviously is a subgroup (due to lemma) of . We will show, that it is normal. Let and . We have to show, that . Of course
because is normal (here correspond to ). It follows from lemma (note, that is -finite), that
which shows, that is normal. To obtain the contradiction, we need to show, that there exists with at least elements, such that is nontrivial and proper (because in this case is simple).
Let be such that and let be such that . Let be any -finite and -finite subset of with at least elements (such subset exists). Then is nontrivial, because is nontrivial.
Now assume, that , i.e. assume, that there exists with , such that . Then (due to lemma) is -finite, and since we have that for any the following holds:
On the other hand, for we have . This shows, that , but and . Contradiction. Thus , so is proper.
This completes the proof.
Remark. This proposition shows, that the class of simple groups is actually a proper class, i.e. it is not a set. Therefore studying infinite simple groups can be very difficult.
|Title||examples of infinite simple groups|
|Date of creation||2013-03-22 19:09:17|
|Last modified on||2013-03-22 19:09:17|
|Last modified by||joking (16130)|