Analogue of critical subgroup theorem for infinite abelian-by-nilpotent p-groups
Suppose is a prime number and is an infinite P-group (?) that is Abelian-by-nilpotent group (?): has an abelian normal subgroup such that the quotient is a nilpotent group. Then, has a characteristic subgroup satisfying the following properties:
- , viz., the Frattini subgroup is contained inside the center (i.e., is a Frattini-in-center group (?)).
- (i.e., is a Commutator-in-center subgroup (?) of ).
- (i.e., is a Self-centralizing subgroup (?) of ).
- is coprime automorphism-faithful in : If is a non-identity automorphism of such that the order of is relatively prime to , then the restriction of to is a non-identity automorphism of .
This a the generalization of a critical subgroup to a possibly infinite p-group.
- Thompson's critical subgroup theorem: Thompson's critical subgroup theorem is the version of this result for finite groups.
- Equivalence of definitions of abelian-by-nilpotent group: This states that is abelian-by-nilpotent if and only if some member of its lower central series is abelian; in particular, that member of the lower central series is an abelian characteristic subgroup with a nilpotent quotient group.
- Every abelian characteristic subgroup is contained in a maximal among abelian characteristic subgroups
- Characteristic implies normal
- Third isomorphism theorem
- Nilpotence is quotient-closed
The proof of the result is exactly the same as for finite groups. The main tricky first step is to show that there exists a subgroup that is Maximal among abelian characteristic subgroups (?), and such that the quotient is a nilpotent group. We do this first step in a separate subsection.
There exists a subgroup maximal among abelian characteristic subgroups with a nilpotent quotient group
Given: A group with an abelian normal subgroup such that is nilpotent.
To prove: There exists a subgroup of such that is nilpotent and is maximal among abelian characteristic subgroups.
- There exists an abelian characteristic subgroup of such that is nilpotent: This is a consequence of fact (1).
- is contained in a subgroup that is maximal among abelian characteristic subgroups: This follows from fact (2).
- is abelian characteristic and is abelian: By fact (3), both and are normal in , and by fact (4), we have . Thus, is isomorphic to a quotient of a nilpotent group. By fact (5), is nilpotent.