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

MINIMAL NONMODULAR FINITE p-GROUPS

Zvonimir Janko

Abstract.   We describe first the structure of finite minimal nonmodular 2-groups G. We show that in case |G| > 2⁵, each proper subgroup of G is Q₈-free and G/(G) is minimal nonabelian of order 2⁴ or 2⁵. If |G/(G)| = 2⁴, then the structure of G is determined up to isomorphism (Propositions 2.4 and 2.5). If |G/(G)| = 2⁵, then Ω₁(G) E₈ and G/Ω₁(G) is metacyclic (Theorem 2.8).

Then we classify finite minimal nonmodular p-groups G with p > 2 and |G| > p⁴ (Theorems 3.5 and 3.7). We show that G/(G) is nonabelian of order p³ and exponent p and (G) is metacyclic. Also, G/Ω₁(G) E_p and G/Ω₁(G) is metacyclic.

2000 Mathematics Subject Classification.   20D15.

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