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On the Semisimplicity of Modular Group Algebras. II

Published online by Cambridge University Press:  20 November 2018

D. S. Passman
D. S. Passman*
Affiliation:
Yale University, New Haven, Connecticut
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Let G be a discrete group, let Kbe an algebraically closed field of characteristic p > 0 and let KGdenote the group algebra of Gover K.In a previous paper (2) I studied the Jacobson radical JKGof KGfor groups Gwith big abelian subgroups or quotient groups. It is therefore natural to next consider metabelian groups, and I do this here. The main result is as follows.

THEOREM 1. Let K be an algebraically closed field of characteristic p and let a group G have a normal abelian subgroup A with G/A abelian. Then JKG ≠ {0} if and only if G has an element g of order p such that the A-conjugacy class gA is finite and such that the group is periodic.

Note that since and G/Ais abelian, we do in fact have .

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1137 - 1145
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Passman, D. S., Nil ideals in group rings, Michigan Math. J. 9 (1962), 375–384.Google Scholar
2. Passman, D. S., On the semisimplicity of modular group algebras, Proc. Amer. Math. Soc. 20 (1969), 515–519.Google Scholar
3. Scott, W. R., Group theory (Prentice-Hall, Englewood Cliffs, N.J., 1964).Google Scholar