Symmetric groups

symmetric groups: G = Sn. (i) Let g1, g2 belong G be two disjoint cycles, and let g = g1g2. Prove that o(g) = lcm { o( g1), o(g2)}, where lcm stands for the least common multiple. (ii) Let g= g1g2 ... gr belong G, where g1,g2, ... gr are disjoint cycles. Prove that o(g) = lcm {o(g1), o(g2), ... o(gr)}. Can you tell me how to start, and step by step guide?

Isomorphism

This week lecture is taught about Isomorphism, automorphism and Inner automorphism, but I don't understand what are they. I only know the symbol Aut( ) , Inn() , ~ on = , bijection Can you give some simple examples?

Ring Theory/Largest two-sided ideal

Let I be a right ideal of a ring R and let A = {r in R: (R/I)r = 0}. Prove that A is the largest two-sided ideal of R contained in I.

Ring Theory/Nil radical

Let R be a ring with the property that every element is either nilpotent or invertible. If a, b, c are in R with a and b nilpotent, show that ac, ca, and a + b are nilpotent. For the latter, first observe that a + b cannot equal 1. Conclude that Nil (R) is the set of all nilpotent elements of R. (nil radical Nil (R) is defined to be the sum of all nil two-sided ideals of R)

Prove that if G is a group of order n and F is any field then GLn(F) contains

Prove that if G is a group of order n and F is any field then GLn(F) contains a subgroup isomorphic to G.