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.

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从群论观点看 Steenrod 代数

在论文“The Steenrod algebra and its dual”中，J.Milnor 确定了对偶 Steenrod 代数的结构，它是有限类型的分级交换 Hopf 代数。我们考虑由模 $p$ Steenrod 代数的对偶 Hopf 代数表示的仿射群方案 $G_p$。然后，$G_p$ 将有限特征$p$ 的素域上的分级交换代数$A_*$ 分配给$A_*$ 上的加法形式群律的一组同构，其群结构由以下组合给出正式的电源系列。本文的目的是利用 $G_p(A_*)$ 的这种表示来展示 $G_p$ 的一些群论性质。我们给出了 $G_p$ 的子群方案的递减过滤，我们用它来估计 $G_p$ 的有限子群方案的下中心序列的长度。