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If the action of $S$ on $E$ is free, we show that the skew group algebra $\\Lambda G$ and $\\Lambda$ have the same finitistic dimension, and have the same strong global dimension if the fixed algebra $\\Lambda^S$ is a direct summand of the $\\Lambda^S$-bimodule $\\Lambda$. 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Under the assumption that $\\Lambda$ has a complete set $E$ of primitive orthogonal idempotents, closed under the action of a Sylow $p$-subgroup $S \\leqslant G$. If the action of $S$ on $E$ is free, we show that the skew group algebra $\\Lambda G$ and $\\Lambda$ have the same finitistic dimension, and have the same strong global dimension if the fixed algebra $\\Lambda^S$ is a direct summand of the $\\Lambda^S$-bimodule $\\Lambda$. 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