Subgroup-defining function value is characteristic
From Groupprops
Contents
Definition
Suppose is a subgroup-defining function, i.e.,
associates to every group
a subgroup
, with the property that given an isomorphism
, the image of
under
is
.
Then, for every group ,
is a characteristic subgroup of
.
Related facts
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 |
Proof
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.