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


Zvonimir Janko

Mathematical Institute, University of Heidelberg, 69120 Heidelberg, Germany

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/mho2(G) is minimal nonabelian of order 24 or 25. If |G/mho2(G)| = 24, then the structure of G is determined up to isomorphism (Propositions 2.4 and 2.5). If |G/mho2(G)| = 25, then Ω1(G) = E8 and G1(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/mho1(G) is nonabelian of order p3 and exponent p and mho1(G) is metacyclic. Also, G1(G) = Ep3 and G1(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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