Generators for the Mod-p Cohomology of the Steinberg Summand of Thom Spectra Over $\mathrm {B}(\mathbb {Z}/p)^{n}$-Odd Primary Cases

For a given odd prime number p, in this paper we construct a minimal generating set for the mod-p cohomology of the Steinberg summand of a family of Thom spectra over the classifying space of an elementary p-abelian group. This resolves the remaining cases for odd prime numbers of a problem studied previously by M. Inoue (Contemporary Mathematics, vol. 293, pp. 125–139, 2002) and (J. Lond. Math. Soc. 75: 317–329, 2007) and by the author (J. Algebra 381: 164–175, 2013).