Vipul: New page: {{group metaproperty satisfaction| property = simple group| metaproperty = directed union-closed group property}} ==Statement== ===Verbal statement=== A directed union of simple subgrou...

2008-09-07T17:33:07Z

New page: {{group metaproperty satisfaction| property = simple group| metaproperty = directed union-closed group property}} ==Statement== ===Verbal statement=== A directed union of simple subgrou...

New page

{{group metaproperty satisfaction|
property = simple group|
metaproperty = directed union-closed group property}}

==Statement==

===Verbal statement===

A directed union of simple subgroups is simple.

===Statement with symbols===

Suppose <math>G</math> is a group, <math>I</math> is a nonempty directed set, and <math>H_i, i \in I</math> is a collection of subgroups such that <math>i < j \implies H_i \le H_j</math>. Then, if all the <math>H_i</math>s are simple, so is their union.

==Definitions used==

===Directed set===

A directed set is a partially ordered set <math>(I, \le)</math> such that if <math>i,j \in I</math>, there exists <math>k \in I</math>, such that <math>i \le k, j \le k</math>.

===Simple group===

A nontrivial group is simple if it has no proper nontrivial [[normal subgroup]].

==Facts used==

# [[Directed union of subgroups is subgroup]]
# [[Normality satisfies transfer condition]]: The intersection of a normal subgroup with another subgroup is normal in that subgroup.

==Proof==

'''Given''': A group <math>G</math>, a collection of simple subgroups <math>H_i</math> indexed by a directed set <math>I</math>, such that <math>i < j \implies H_i \le H_j</math>.

'''To prove''': The union of <math>H_i</math> is a simple subgroup.

'''Proof''': Suppose <math>H</math> is the union of the <math>H_i</math>s. By fact (1), <math>H</math> is a subgroup, so it suffices to show that <math>H</math> is simple. We do this by showing that any normal subgroup <math>N</math> of <math>H</math> is either trivial or equal to <math>H</math>.

For each <math>i</math>, consider <math>N \cap H_i</math>. By fact (2), this is a normal subgroup of <math>H_i</math>, hence is either trivial or equals the whole of <math>H_i</math>.
Suppose <math>N \cap H_i = H_i</math>. Then, <math>N</math> is a normal subgroup of <math>H</math> containing <math>H_i</math>.

Now, consider any subgroup <math>H_j, j \ne i</math>. By the definition of directed set, there exists <math>k \in I</math> such that <math>i,j < k</math>. So, <math>H_k</math> contains both <math>H_i</math> and <math>H_j</math>. Thus, the intersection <math>N \cap H_k</math> is nontrivial (since the intersection contains <math>H_i</math>). By the simplicity of <math>H_k</math>, and fact (2) again, <math>N \cap H_k = H_k</math>, so <math>H_k \le N</math>. In particular, <math>H_j \le N</math>. Thus, ''every'' <math>H_j, j \in I</math>, is contained in <math>N</math>, Thus, <math>N = H</math>.

Thus, if <math>N \cap H_i = H_i</math> for any <math>i \in I</math>, <math>N = H</math>. This leaves the case that <math>N \cap H_i</math> is trivial for every <math>i \in I</math>, forcing <math>N \cap (\bigcup_i) H_i</math> to be trivial, and thus forcing <math>N \cap H = N</math> to be trivial.

Simplicity is directed union-closed - Revision history

Vipul: New page: {{group metaproperty satisfaction| property = simple group| metaproperty = directed union-closed group property}} ==Statement== ===Verbal statement=== A directed union of simple subgrou...