Isomorphism

This week lecture is taught about Isomorphism, automorphism and Inner automorphism, but I don't understand what they are. 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.

Group of order 9

This is the question: Consider small groups. (i) Show that a group of order 9 is isomorphic to Z9 or Z3 x Z3 (ii) List all groups of order at most 10 (up to isomorphism)