Let K be the operator in Solve the following equations or e

Let K be the operator in (*). Solve the following equations, or explain why there is no solution. Ku - 2u = x(pi-x) \"Ku(x) = integral_0^pi k (x, y) u(y) dy where K (x, y) = y (pi - x) if x > y and K (x, y) = x (pi - y) if x

Solution

Ans-

The main dierence b etween these two groups is that

Z

4

has elements of order

four, while

Z

2

Z

2

do es not. Thus, if we have a group

G

of order four, it is

isomorphic to either

Z

4

or

Z

2

Z

2

, and we can gure out which one by either:

1. Showing that the order of every element in

G

is less than or equal to two

(so

G

Z

2

Z

2

), or

2. Showing that at least one element of

G

has order four (so

G

Z

4

).

In this problem, we have an Ab elian group

G

=

Z

4

Z

4

and two (normal)

subgroups

H

=

f

(0

;

0)

;

(2

;

0)

;

(0

;

2)

;

(2

;

2)

g

and

K

=

h

(1

;

2)

i

=

f

(0

;

0)

;

(1

;

2)

;

(2

;

0)

;

(3

;

2)

g

:

We will lo ok at the factor groups

G=H

and

G=K

and determine if they are

isomorphic to either

Z

4

or

Z

2

Z

2

. Note that the order of b oth

G=H

and

G=K

is 4, since

G

itself has order 16 and b oth

H

and

K

have order 4, so

j

G=H

j

=

j

G

j

j

H

j

=

4

4

4

= 4