Abelian-to-normal replacement theorem for prime-square index

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This article defines a replacement theorem
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Statement

Let p be any prime number (possibly p = 2). Then, if P is a finite p-group (i.e., a group of prime power order where the underlying prime is p) and A is an abelian subgroup of index p^2 in P, then there is an abelian normal subgroup of P of index p^2. Moreover, we can choose this abelian normal subgroup so that it is contained in the normal closure of A in P.

Related facts

Stronger facts

Facts used

  1. Congruence condition on number of abelian subgroups of prime index

Proof

Given: A finite p-group P for some prime p, an abelian subgroup A of P of index p^2.

To prove: There exists an abelian normal subgroup B of P of index p^2, contained inside the normal closure of A in P.

Proof:

  1. If A is normal in P, we are done. Otherwise, A is a 2-subnormal subgroup and its normal closure is a maximal subgroup M of P containing A. M is normal and has index p in P, and A is normal and has index p in M.
  2. By fact (1), the number of abelian subgroups of M is congruent to 1 modulo p.
  3. Since M is normal in P, P acts on M by conjugation and the orbits of abelian maximal subgroups are therefore of size equal to a power of p. Since the total number of abelian maximal subgroups is 1 modulo p, there exists an orbit of size 1, and the member of this orbit is thus an abelian normal subgroup of index p.

References

Journal references