Glasnik Matematicki, Vol. 46, No.1 (2011), 1-5.

FINITE p-GROUPS G WITH p>2 AND \rm d(G)>2 HAVING EXACTLY ONE MAXIMAL SUBGROUP WHICH IS NEITHER ABELIAN NOR MINIMAL NONABELIAN

Zvonimir Janko

Abstract.   We give here a complete classification (up to isomorphism) of the title groups (Theorems 1, 3 and 5). The corresponding problem for p=2 was solved in [4] and for p>2 with d(G)=2 was solved in [5]. This gives a complete solution of the problem Nr. 861 of Y. Berkovich stated in [2].

2000 Mathematics Subject Classification.   20D15.

Key words and phrases.   Minimal nonabelian p-groups, A₂-groups, metacyclic p-groups, Frattini subgroups, Hall-Petrescu formula, generators and relations, congruences mod p.

DOI: 10.3336/gm.46.1.11

References:

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

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

3. Y. Berkovich and Z. Janko, Groups of prime power order. Vol. 3, Walter de Gruyter, Berlin-New York, to appear 2011.

4. Z. Božikov and Z. Janko, Finite 2-groups with exactly one maximal subgroup which is neither abelian nor minimal nonabelian, Glas. Mat. Ser. III 45(65) (2010), 63-83.
   MathSciNet     CrossRef

5. Z. Janko, Finite p-groups G with p>2 and d(G)=2 having exactly one maximal subgroup which is neither abelian nor minimal nonabelian, Glas. Mat. Ser. III 45(65) (2010), 441-452.
   MathSciNet     CrossRef