Hello i need help with modern algebra
I've been busy doing hardwork on modern Algebra On "group and ring
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
4.Let G be a group of order 60,and let S be the set of Sylow
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
|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?.....