#### Glasnik Matematicki, Vol. 39, No.2 (2004), 221-233.

### MINIMAL NONMODULAR FINITE *p*-GROUPS

### Zvonimir Janko

Mathematical Institute, University of Heidelberg,
69120 Heidelberg, Germany

*e-mail:* `janko@mathi.uni-heidelberg.de`

**Abstract.** We describe first the structure of finite
minimal nonmodular 2-groups *G*. We show that in case
|*G*| > 2^{5}, each proper subgroup
of *G* is *Q*_{8}-free
and *G*/(*G*)
is minimal nonabelian of order 2^{4} or 2^{5}.
If |*G*/(*G*)| =
2^{4}, then the structure
of *G* is determined up to isomorphism (Propositions 2.4 and 2.5).
If |*G*/(*G*)| =
2^{5}, then
Ω_{1}(*G*)
*E*_{8} and
*G*/Ω_{1}(*G*)
is metacyclic (Theorem 2.8).

Then we classify finite minimal nonmodular
*p*-groups *G* with *p* > 2 and
|*G*| > *p*^{4} (Theorems 3.5 and 3.7).
We show that
*G*/(*G*)
is nonabelian of order *p*^{3} and
exponent *p* and
(*G*)
is metacyclic. Also,
*G*/Ω_{1}(*G*)
*E*_{p}
and
*G*/Ω_{1}(*G*)
is metacyclic.

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

**Key words and phrases.** *p*-group, modular group,
nonmodular group, quaternion group, minimal nonabelian group.

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