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Journal of the London Mathematical Society 1991 s2-43(1):23-36; doi:10.1112/jlms/s2-43.1.23
© 1991 by London Mathematical Society
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© Oxford University Press

Lie Dimension Subgroups, Lie Nilpotency Indices, and the Exponent of the Group of Normalized Units

Aner Shalev

The Hebrew University of Jerusalem Jerusalem 91904, Israel

Let K be a field of characteristic p > 0, and let G be a finite p-group. Let U be the group of normalized units of the modular group algebra KG. In this paper we study the relation between exp (U) and exp (G). The main result shows that, if p ≥ 7 and exp(G)3 > |G|, then G and U have the same exponent. We also show that, in general, exp(U) cannot be bounded above by any fixed function of exp(G).

The method involves a reduction to problems in Lie nilpotency indices, which are solved via an extensive study of Lie dimension subgroups. Some results for smaller p are also given.


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