The Steenrod algebra from the group theoretical viewpoint

Abstract

In the paper "The Steenrod algebra and its dual", J.Milnor determined the structure of the dual Steenrod algebra which is a graded commutative Hopf algebra of finite type. We consider the affine group scheme $G_p$ represented by the dual Hopf algebra of the mod $p$ Steenrod algebra. Then, $G_p$ assigns a graded commutative algebra $A_*$ over a prime field of finite characteristic $p$ to a set of isomorphisms of the additive formal group law over $A_*$, whose group structure is given by the composition of formal power series. The aim of this paper is to show some group theoretic properties of $G_p$ by making use of this presentation of $G_p(A_*)$. We give a decreasing filtration of subgroup schemes of $G_p$ which we use for estimating the length of the lower central series of finite subgroup schemes of $G_p$. We also give a successive quotient maps $G_p\xrightarrow{\rho_0}G_p^{\langle1\rangle}\xrightarrow{\rho_1}G_p^{\langle2\rangle}\xrightarrow{\rho_2}\cdots\xrightarrow{\rho_{k-1}} G_p^{\langle k\rangle}\xrightarrow{\rho_k}G_p^{\langle k+1\rangle}\xrightarrow{\rho_{k+1}}\cdots$ of affine group schemes over a prime field ${\boldsymbol F}_p$ such that the kernel of $\rho_k$ is a maximal abelian subgroup.