Yakov Berkovich

Groups of prime power order

This is the fifth volume of a comprehensive and elementary treatment of finite p-group theory. Topics covered in this volume include theory of linear algebras and Lie algebras. The book contains many dozens of original exercises (with difficult exercises being solved) and a list of about 900 research problems and themes.

Groups with few characters of small degrees

Letd>1 be a proper divisor of the order of a finite groupG and let σd(G) be the sum of squares of degrees of those irreducible characters whose degrees are not divisible byd. It is easy to see thatd divides σd(G). The groupsG such that σd(G) =d coincide with Frobenius groups whose kernel has indexd (see G. Karpilovsky,Group Representations, Volume 1, Part B, North-Holland, Amsterdam, 1992, Theorem 37.5.5). In this note we study the case σd(G) = 2d in some detail. In particular, ifG is a 2-group, it is of maximal class (Remark 3(b)).

ON

The isomorphism types of all minimal nonabelian subgroups (

\mathcal A_1

= \mathcal A_1

- AND

-subgroups) of

\mathcal A_2

\mathcal A_2

-SUBGROUPS OF FINITE p-GROUPS

-groups are described. This allows us to classify the nonabelian p-groups, p > 2, that have no p isomorphic

Finite groups all of whose small subgroups are normal

\mathcal A_1

On the number of minimal nonabelian subgroups in a nonabelian finite p-group

-subgroups of minimal order. In particular, if a p-group G is neither abelian nor

Some remarks on subgroups, epimorphic images and p′-automorphisms of finite p-groups

\mathcal A_1