Glasnik Matematicki, Vol. 41, No.2 (2006), 239-258.

SHORT PROOFS OF SOME BASIC CHARACTERIZATION THEOREMS OF FINITE p-GROUP THEORY

Yakov Berkovich

Abstract.   We offer short proofs of such basic results of finite p-group theory as theorems of Blackburn, Huppert, Ito-Ohara, Janko, Taussky. All proofs of those theorems are based on the following result: If G is a nonabelian metacyclic p-group and R is a proper G-invariant subgroup of G', then G/R is not metacyclic. In the second part we use Blackburn's theory of p-groups of maximal class. Here we prove that a p-group G is of maximal class if and only if Ω₂*(G) = 〈 x ∈ G | o(x) = p² 〉 is of maximal class. We also show that a noncyclic p-group G of exponent > p contains two distinct maximal cyclic subgroups A and B of orders > p such that |A ∩ B| = p, unless p = 2 and G is dihedral.

2000 Mathematics Subject Classification.   20D15.

Key words and phrases.   Finite p-groups, metacyclic p-groups, minimal nonabelian p-groups, p-groups of maximal class, regular and absolutely regular p-groups, powerful p-groups.

DOI: 10.3336/gm.41.2.07

References:

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

2. Y. Berkovich, Alternate proofs of some basic theorems of finite group theory, Glas. Mat. Ser. III 40(60) (2005), 207-233.
   MathSciNet

3. Y. Berkovich, Groups of Prime Power Order, Part I, in preparation.

4. Y. Berkovich, Alternate proofs of two theorems of Philip Hall on finite p-groups, and some related results, J. Algebra 294 (2005), 463-477.
   MathSciNet     CrossRef

5. Y. Berkovich, On the metacyclic epimorphic images of finite p-groups, Glas. Mat. Ser. III 41(61) (2006), 259-269.
   MathSciNet CrossRef

6. Y. Berkovich, Finite p-groups with few minimal nonabelian subgroups. With an appendix by Z. Janko, J. Algebra 297 (2006), 62-100.
   MathSciNet     CrossRef

7. Y. Berkovich and Z. Janko, Groups of Prime Power Order, Part II, in preparation.

8. Y. Berkovich and Z. Janko, Structure of finite $p$-groups with given subgroups, in: Contemp. Math. 402, Amer. Math. Soc., Providence, 2006, 13-93.
   MathSciNet

9. Ya. G. Berkovich and E. M. Zhmud, Characters of Finite Groups, Part 1, American Mathematical Society, Providence, 1998.
   MathSciNet

10. N. Blackburn, On prime-power groups with two generators, Proc. Cambridge Philos. Soc. 54 (1958), 327-337.
    MathSciNet

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

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

13. B. Huppert, Über das Produkt von paarweise vertauschbaren zyklischen Gruppen, Math. Z. 58 (1953), 243-264.
    MathSciNet     CrossRef

14. I. M. Isaacs, Character Theory of Finite Groups, Academic Press, New York-London, 1976.
    MathSciNet

15. N. Ito and A. Ohara, Sur les groupes factorisables par deux 2-groupes cycliques I, II, Proc. Japan Acad. 32 (1956), 736-743.
    MathSciNet

16. Z. Janko, Finite 2-groups with no normal elementary abelian subgroups of order 8, J. Algebra 246 (2001), 951-961.
    MathSciNet     CrossRef

17. Z. Janko, On maximal cyclic subgroups in finite p-groups, Math. Z. 254 (2006), 29-31.
    MathSciNet     CrossRef

18. Z. Janko, Finite 2-groups with exactly four cyclic subgroups of order 2ⁿ, J. Reine Angew. Math. 566 (2004), 135-181.
    MathSciNet     CrossRef

19. Z. Janko, Finite 2-groups G with Ω₂*(G) metacyclic, Glas. Mat. Ser. III 41(61) (2006), 71-76.
    CrossRef

20. Z. Janko, Finite 2-groups with exactly one nonmetacyclic maximal subgroup, submitted.

21. A. Lubotzky and A. Mann, Powerful p-groups. I. Finite groups, J. Algebra 105 (1987), 484-505.
    MathSciNet     CrossRef

22. L. Redei, Das "schiefe Produkt" in der Gruppentheorie mit Anwendung auf die endlichen nichtkommutativen Gruppen mit lauter kommutativen echten Untergruppen und die Ordnungszahlen, zu denen nur kommutative Gruppen gehören, Comment. Math. Helv. 20 (1947), 225-264.
    MathSciNet