Glasnik Matematicki, Vol. 39, No.2 (2004), 213-220.

ON A LEMMA OF THOMPSON

Yakov Berkovich

Abstract.   In Theorem 3 we improve a lemma of Thompson omitting one of its conditions. In that lemma the structure of T, a Sylow 2-subgroup of G, is described only. In contrast to that lemma, we describe in detail the structure of the whole group G and embedding of T in G. In Theorem 4 we consider a similar, but more general, situation for groups of odd order.

2000 Mathematics Subject Classification.   20C15.

Key words and phrases.   Solvable, extraspecial, special and metacyclic p-groups, Blackburn's theorem, the stabilizer of a chain.

DOI: 10.3336/gm.39.2.03

References:

1. Y. Berkovich, On abelian subgroups of p-groups, J. Algebra 199 (1998), 262-280.

2. Y. G. Berkovich and E. M. Zhmud, Characters of Finite Groups. Part 1, Translations of Mathematical Monographs 172, American Mathematical Society, Providence, RI, 1997.

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

4. B. Huppert, Endliche Gruppen, Band I, Springer, Berlin, 1967.

5. I. M. Isaacs, Algebra, a graduate course, Brooks/Cole, Pacific Grove, 1993.

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

7. M. Suzuki, Group Theory I, Springer, Berlin, 1982.

8. J. G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable I, Bull. Amer. Math. Soc. 74 (1968), 383-437.