Generators for the mod 2 cohomology of the Steinberg summand of Thom spectra over B(Z/2)n

Let Ln,k be the Steinberg summand of the ideal generated by the k-th power of the top Dickson class of H⁎(B(Z/2)n;Z/2). The module Ln,k is the mod 2 cohomology of the Steinberg summand of a Thom spectrum over the classifying space B(Z/2)n. In this paper, using the Kameko map (Kameko, 1990 [6], ) and short exact sequences induced by the cofibration sequences constructed by Takayasu (1999) [15], , we construct a minimal generating set for Ln,k as a module over the mod 2 Steenrod algebra. This generalises the result of Masateru Inoue (2002) [5] for the cases k=0,1.