Subgroup-defining function value is characteristic
Then, for every group , is a characteristic subgroup of .
Conclusion about normality
Combined with the fact that characteristic implies normal, this tells us that any subgroup-defining function value is normal.
Applications to specific subgroup-defining functions
|Subgroup-defining function||Meaning||Proof that it is characteristic||Proof of stronger properties satisfied for the particular subgroup-defining function|
|center||elements that commute with everything||center is characteristic||center is bound-word, center is strictly characteristic, center is direct power-closed characteristic|
|derived subgroup||products of commutators||derived subgroup is characteristic||derived subgroup is verbal, derived subgroup is fully invariant|
|Frattini subgroup||intersection of all maximal subgroups||Frattini subgroup is characteristic|
Given: A group , a subgroup-defining function , an automorphism of .
To prove: .
Proof: Since is an automorphism, it is in particular an isomorphism. Thus, the definition of subgroup-defining function above, setting , gives us what we need to prove.