MathStruggle
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- Sep 29, 2016
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Q on Abstract Algebra (1): Let α = (x0 · · · xr−1) be an r-cycle.
(a) Let α = (x0 · · · xr−1) be an r-cycle. In lecture, we have shown that for everypositive integers k and i, αk(x0) = xk, where the subscript is read mod r. Now,show that r is the smallest positive integer such that αr = (1).
(b) If α ∈ Sn, show that there exists a positive integer k such that αk = (1).
(c) If α ∈ Sn, show that α−1 is a nonnegative power of α.
(a) Let α = (x0 · · · xr−1) be an r-cycle. In lecture, we have shown that for everypositive integers k and i, αk(x0) = xk, where the subscript is read mod r. Now,show that r is the smallest positive integer such that αr = (1).
(b) If α ∈ Sn, show that there exists a positive integer k such that αk = (1).
(c) If α ∈ Sn, show that α−1 is a nonnegative power of α.