Analogue of critical subgroup theorem for infinite abelian-by-nilpotent p-groups

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Statement

Suppose p is a prime number and G is an infinite P-group (?) that is Abelian-by-nilpotent group (?): G has an abelian normal subgroup N such that the quotient G/N is a nilpotent group. Then, G has a characteristic subgroup H satisfying the following properties:

  1. \Phi(H) \le Z(H), viz., the Frattini subgroup is contained inside the center (i.e., H is a Frattini-in-center group (?)).
  2. [G,H] \le Z(H) (i.e., H is a Commutator-in-center subgroup (?) of G).
  3. C_G(H)= Z(H) (i.e., H is a Self-centralizing subgroup (?) of G).
  4. H is coprime automorphism-faithful in G: If \sigma is a non-identity automorphism of G such that the order of \sigma is relatively prime to p, then the restriction of \sigma to H is a non-identity automorphism of H.

This a the generalization of a critical subgroup to a possibly infinite p-group.

Related facts

Facts used

  1. Equivalence of definitions of abelian-by-nilpotent group: This states that G 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.
  2. Every abelian characteristic subgroup is contained in a maximal among abelian characteristic subgroups
  3. Characteristic implies normal
  4. Third isomorphism theorem
  5. Nilpotence is quotient-closed

Proof

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 G with an abelian normal subgroup N such that G/N is nilpotent.

To prove: There exists a subgroup K of G such that G/K is nilpotent and K is maximal among abelian characteristic subgroups.

Proof:

  1. There exists an abelian characteristic subgroup L of G such that G/L is nilpotent: This is a consequence of fact (1).
  2. L is contained in a subgroup K that is maximal among abelian characteristic subgroups: This follows from fact (2).
  3. K is abelian characteristic and G/K is abelian: By fact (3), both K and L are normal in G, and by fact (4), we have (G/L)/(K/L) \cong G/K. Thus, G/K is isomorphic to a quotient of a nilpotent group. By fact (5), G/K is nilpotent.

Construction of the critical subgroup

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