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Published online by Cambridge University Press: 24 October 2008
If F is a (commutative) field let XF denote the class of all groups G such that every irreducible FG module has finite dimension over F. In the first paper (9) of this series we classified finitely generated soluble XF-groups for each field F and in the second (10) we characterized soluble XF-groups for each field F of characteristic zero. Here we consider soluble XF-groups over fields F of positive characteristic.