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.