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Albegra 1 Questions
Hey my final is tommorow and I'm having trouble with a few of the practice questions my teacher has listed.
Let sigma = sigma1 (dot) sigma2 (dot) .... sigma(n) in Sn be a product of disjoint commuting cycles whose orders are o(sigma(i)) = ni - Explain why sigma^k = e <=> sigma(i)^k = e for each 1 =< i =< r - assuming the above is true, prove that the order of sigma is he least common multiple l = lcm(n1,...., nr) And the second question is Consider the automorphism group G = Aut(Z25m +) congruent (U25m, dot) -what is the order of G and its p-sylow subgroups? -Show that G is a direct product AXB of two familiar groups. Identify A and B, explicitly, ip to isomorphism -Up to ismoophism, what the 2-sylow and 5-sylow subgroups in G and how many are there of each type? I figured to try the math forum if you guys can give me a hand that'd be great. |
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