Hello i need help with modern algebra

Dear All

I've been busy doing hardwork on modern Algebra On "group and ring

theory".

It's seem new to me! i try to read the book "Contemporary Abstract

Algebra: by Joseph A. Gallian".

I try to solve these following questions(I think it may be simple for

people that already took this course befor).

The questions are in the following ordered:

1. Suppose O:G--->H is a homomorphism.(for the O notation it actually

Thih but i can't write in correct notation due to board).

Suppose that K<|G with K |^| Ker(O)=1 .Show that K is isomorphic to a

subgroup of H.( where <| stand for normal subgroup,,,,and |^| mean

intersect (Khmer word Prosob.)).

2.Consider a rectangle box of size a * a * b where a # b.(* stand for

multiply,,,and # stand for Not equal to).Describe the rotational

symmetry group of this box (it is isomorphic to the dihedral group),and

use Burnside's theorem to find the number of rotationally distinct ways

of colouring the edges using three colours.

3.Assume G/Z(G) is cyclic.show that G = <x> Z(G) for some x belong to G

and hence show that G would have to be Albelian.This is ofcourse a

contradiction,proving that it is not in fact possible for G/Z(G) to be

cyclic.

4.Let G be a group of order 60,and let S be the set of Sylow

3_subgroups.

Define the action of G on S by conjugacy and explain how it can be used

to generate a group homomorphism Phi:G--->Sn where n=|s|.

Explaine Why Im(Phi)#1.

(phi actually(phi notation that phi=3.1416...) Sn i mean S sub n and

ofcourse

|s| is absolute value of s,,,,# stand for not equal to).

5.Use the result from question 4 to show that a simple group of order

60 is isomorphic to A5.

Any idear Anyone?.....

Kind Reguard

Pisey Guy