#### Glasnik Matematicki, Vol. 44, No.1 (2009), 167-175.

### FINITE *p*-GROUPS IN WHICH SOME SUBGROUPS ARE
GENERATED BY ELEMENTS OF ORDER *p*

### Yakov Berkovich

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

**Abstract.** We prove that if a *p*-group *G* of exponent
*p*^{e} > *p*
has no subgroup *H* such that
|Ω_{1}(*H*)| = *p*^{p} and
*H*/Ω_{1}(*H*) is cyclic of order
*p*^{e-1} ≥ *p* and *H* is
regular provided *e* = 2, then *G* is either absolutely regular or of
maximal class. This result supplements the fundamental theorem of
Blackburn on *p*-groups without normal subgroups of order *p*^{p} and
exponent *p*. For *p* > 2, we deduce even
stronger result than (respective result for *p* = 2 is unknown) a theorem of Bozikov and Janko.

**2000 Mathematics Subject Classification.**
20D15.

**Key words and phrases.** *p*-groups of maximal class, regular and absolutely
regular *p*-groups, metacyclic *p*-groups, L_{p}-groups.

**Full text (PDF)** (free access)
DOI: 10.3336/gm.44.1.09

**References:**

- Y. Berkovich,
*On subgroups and epimorphic images of finite
**p*-groups, J. Algebra **248** (2002), 472-553.

MathSciNet
CrossRef

- Y. Berkovich,
Groups of Prime Power Order, volume 1, Walter
de Gruyter, 2008.

MathSciNet

- Y. Berkovich,
*A generalization of theorems of Ph. Hall and
N. Blackburn and an application to non-regular **p*-groups, Math. USSR
Izv. **35** (1971), 815-844.

MathSciNet
CrossRef

- Y. Berkovich and Z. Janko,
*Structure of finite $p$-groups with
given subgroups*, Contemp. Math. **402** (2006), 13-93.

MathSciNet

- N. Blackburn,
*Generalization of certain elementary theorems
on **p*-groups, Proc. London Math. Soc. (3) **11** (1961), 1-22.

MathSciNet
CrossRef

- N. Blackburn,
*On a special class of **p*-groups, Acta Math. **100** (1958), 45-92.

MathSciNet
CrossRef

- N. Blackburn,
*Note on a paper of Berkovich*, J. Algebra
**24** (1973), 323-334.

MathSciNet
CrossRef

- Z. Bozikov and Z. Janko,
*Finite 2-groups all of whose
nonmetacyclic subgroups are generated by involutions*, Arch. Math.
**90** (2008), 14-17.

MathSciNet
CrossRef

- P. Hall,
*A contribution to the theory of groups of prime power order*,
Proc. London Math. Soc. **2**, No. 36 (1933), 29-95.

- B. Huppert,
Endliche Gruppen. I., Springer, Berlin, 1967.

MathSciNet

*Glasnik Matematicki* Home Page