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تسجيل مستخدم جديدInfinitely generated pseudocompact modules for finite groups and Weiss Theorem
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One of the most beautiful results in the integral representation theory of finite groups is a theorem of A. Weiss that detects a permutation $R$-lattice for the finite $p$-group $G$ in terms of the restriction to a normal subgroup $N$ and the $N$-fixed points of the lattice, where $R$ is a finite extension of the $p$-adic integers. Using techniques from relative homological algebra, we generalize Weiss Theorem to the class of infinitely generated pseudocompact lattices for a finite $p$-group, allowing $R$ to be any complete discrete valuation ring in mixed characteristic. A related theorem of Cliff and Weiss is also generalized to this class of modules. The existence of the permutation cover of a pseudocompact module is proved as a special case of a more general result. The permutation cover is explicitly described.
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A space $X$ is called {it selectively pseudocompact} if for each sequence $(U_{n})_{nin mathbb{N}}$ of pairwise disjoint nonempty open subsets of $X$ there is a sequence $(x_{n})_{nin mathbb{N}}$ of points in $X$ such that $cl_X({x_n : n < omega}) se
tminus big(bigcup_{n < omega}U_n big) eq emptyset$ and $x_{n}in U_{n}$, for each $n < omega$. Countably compact space spaces are selectively pseudocompact and every selectively pseudocompact space is pseudocompact. We show, under the assumption of $CH$, that for every positive integer $k > 2$ there exists a topological group whose $k$-th power is countably compact but its $(k+1)$-st power is not selectively pseudocompact. This provides a positive answer to a question posed in cite{gt} in any model of $ZFC+CH$.
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