We describe bialgebras of lower-indexed algebraic Steenrod operations over the
field with p elements, p an odd prime. These go beyond the operations
that can act nontrivially in topology, and their duals are closely related to
algebras of polynomial invariants under subgroups of the general linear groups
that contain the unipotent upper triangular groups. There are significant
differences between these algebras and the analogous one for p=2, in
particular in the nature and consequences of the defining Adem relations.
