Small subalgebras of Steenrod and Morava stabilizer algebras

Let $P(j)$ (resp. $S(n)_{(j)})$ be the subalgebra of the Steenrod algebra $\mathcal{A}_p$ (resp. $n$th Morava stabilizer algebra) generated by reduced powers $\mathcal{P}^{p^i}$, $0\le i\le j$ (resp. $t_i$, $1\le i\le j)$. In this paper we identify the dual $P(j-1)^*$ of $P(j-1)$ (resp. $S(n)_{(j)}$, for $j\le n)$ with some Frobenius kernel (resp. $F_{p^n}$-points) of a unipotent subgroup $G(j+1)$ of the general linear algebraic group $GL_{j+1}$. Using these facts, we get the additive structure of $H^*(P(1))=\operatorname{Ext}_{P(1)}(Z/p,Z/p)$ for odd primes.