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

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

### Yakov Berkovich

Department of Mathematics,
University of Haifa,
Mount Carmel, Haifa 31905,
Israel

**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.

**Full text (PDF)** (access from subscribing institutions only)
DOI: 10.3336/gm.46.1.10

**References:**

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

MathSciNet

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

MathSciNet

- A. Mann, personal communication.

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

MathSciNet

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