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- W2009713369 abstract "Let $d$ and $h$ be integers such that either $d geq 2$ and $h = {2^d} - 1$, or $d = 4$ and $h = 5$. Suppose that the group $mathcal {G}$ contains an elementary-abelian $2$-subgroup ${E_d}$ of rank $d$ with an element $sigma$ of order $h$ in its normalizer. We show that if $mathcal {G}$ admits a free and ${{mathbf {F}}_2}$-cohomologically trivial action on ${({S^n})^d}$, then some nontrivial power of $sigma$ centralizes ${E_d}$. The cohomology ring ${H^{ast } }({E_d};{{mathbf {F}}_2}) simeq {{mathbf {F}}_2}[{y_1}, ldots ,{y_d}]$ is a module over the Steenrod algebra $mathcal {A}(2)$. Let $theta in {{mathbf {F}}_2}[{y_1}, ldots ,{y_d}]$, and let $c geq d - 2$ be an integer. We show that $theta$ divides $S{q^{{2^i}}}(theta )$ in the polynomial ring for $0 leq i leq c Leftrightarrow theta = {tau ^{{2^{c - d + 3}}}}pi$ , where $tau$ divides $S{q^{{2^i}}}(tau )$ for $0 leq i leq d - 3$ and $pi$ is a product of linear forms." @default.
- W2009713369 created "2016-06-24" @default.
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- W2009713369 date "1992-01-01" @default.
- W2009713369 modified "2023-09-26" @default.
- W2009713369 title "Subgroup conditions for groups acting freely on products of spheres" @default.
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- W2009713369 doi "https://doi.org/10.1090/s0002-9947-1992-1100700-4" @default.
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