Glasnik Matematicki, Vol. 46, No.1 (2011), 79-101.

THE NUMBER OF SUBGROUPS OF GIVEN ORDER IN A METACYCLIC p-GROUP

Yakov Berkovich

Abstract.   This note was inspired by A. Mann's letter [3] at June 28, 2009, in which the number of subgroups of given order in a metacyclic p-group for odd primes p was computed. Below we present another proof of that result. The offered proof is extended to so called quasi-regular metacyclic 2-groups. In Sec. 2 we compute the number of cyclic subgroups of given order in metacyclic 2-groups. In Sec. 3 we complete computation of the number of subgroups of given order in metacyclic 2-groups. In Sec. 4 we study the metacyclic p-groups with small minimal nonabelian subgroups or sections.

2000 Mathematics Subject Classification.   20D15.

Key words and phrases.   Metacyclic p-groups, quasi-regular metacyclic p-groups, section, Hall's enumeration principle.

DOI: 10.3336/gm.46.1.10

References:

1. Y. Berkovich, Groups of prime power order. Vol. 1, Walter de Gruyter, Berlin, 2008.
   MathSciNet    

2. Y. Berkovich and Z. Janko, Groups of prime power order. Vol. 2, Walter de Gruyter, Berlin, 2008.
   MathSciNet    

3. A. Mann, personal communication.

4. A. Mann, The number of subgroups of metacyclic groups, in: Character theory of finite groups, Contemporary Mathematics 524, AMS, Providence, 2010, 93-95.
   MathSciNet