NORMAL EXTENSIONS OF A CYCLIC p-GROUP

ABSTRACT

The purpose of this paper is to classify the normal extensions G of a cyclic p-group , i.e., , with the irreducible character induced from a faithful linear character of . In fact we determine all normal extensions G of a cyclic p-group by p-subgroups of the automorphism group with the centralizer . Consequently, we have a theorem (Theorem 1) which might characterize the dihedral groups and the generalized quaternion groups of order among non-abelian 2-groups.

ACKNOWLEDGMENT

The author would like to express his sincere gratitude to Professor Claus Michael Ringel, Professor Toshihiko Yamada and Professor Katsusuke Sekiguchi for their helpful suggestions.