View source for Characteristically simple and normal fully normalized implies minimal normal

==Statement==

If <math>H</math> is a [[fact about::normal fully normalized subgroup]] of <math>G</math> and <math>H</math> is [[fact about::characteristically simple group|characteristically simple as a group]], then <math>H</math> is a [[fact about::minimal normal subgroup]] of <math>G</math>.

==Facts used==

# [[Normal upper-hook fully normalized implies characteristic]]: If <math>K \le H \le G</math> are such that <math>K</math> is normal in <math>G</math> and <math>H</math> is fully normalized in <math>G</math>, then <math>K</math> is characteristic in <math>H</math>.

==Proof==

'''Given''': A group <math>G</math>, a characteristically simple, normal and fully normalized subgroup <math>H</math>.

'''To prove''': <math>H</math> is a minimal normal subgroup of <math>G</math>.

'''Proof''': By assumption <math>H</math> is normal, so it suffices to show that any normal subgroup <math>K</math> of <math>G</math> contained in <math>H</math> is either trivial or equal to <math>H</math>. Let's do this.

By fact (1), <math>K</math> is characteristic in <math>H</math>. Since <math>H</math> is characteristically simple, we see that <math>K</math> must be either equal to <math>H</math>, or trivial, completing the proof.