| Abstract: | In 1957, Higman showed that a Lie algebra admitting a fixed-point-free automorphism is nilpotent, and that an analogous result
also holds for a finite soluble group. Two years later, Thompson proved that a finite group having a fixed-point-free automorphism
of prime order is soluble, and consequently nilpotent. Generalizing that situation, a few years ago, Kharchenko set up a conjecture
on the solubility of a Lie algebra L admitting an automorphism of prime order whose fixed points lie in the center of L. A
similar conjecture applies also with finite groups. Here we affirm the latter for the case where the order p of an automorphism
is equal to 2 and deny it for all p>3.
Supported by RFFR grants Nos. 93-01-01501 and 96-01-01893.
Translated fromAlgebra i Logika, Vol. 35, No. 6, pp. 699–708, November–December, 1996. |
