00:01
Here we have to show any homomorphism of additive group q into itself.
00:11
Hence we form x to rx for some r belonging to integer.
00:20
So to prove this we will let phi from the additive group of rational number to itself which is defined by phi of 1 is equal to r and we are considering it to be a homomorphism.
00:43
Now for any natural number n, phi of n can be written as phi of 1 plus 1 so on up to 1 1 is written n times.
00:58
Now since phi is homomorphism, so we can write it as phi of 1, phi of 1 and so on and this will be written n times only.
01:14
So this is written n times or we can write it as phi of 1 is equal to r.
01:23
So this will be r plus r so on up to r.
01:27
Again this is n times so this can be written as rn so from here we get that phi of n is equal to rn now this was the general case if we have phi of 0 so we can write it as phi of n minus n since again phi is homomorphism so we can write it as phi of n into phi of minus n phi of n is equal to rn from here we have this value you plus phi of minus and phi of 0 is basically equal to 0 because phi of 0 is equal to r into 0 which is equal to 0 that means phi of minus n is equal to minus rn.
02:20
So from here here also we have this form now if we consider any which is not equal to 0 and we consider phi of 1 by q plus 1 by q so on up to 1 by q and this is actually written q times q times so this will be basically equal to q into 1 by q which is equal to phi of 1 which is equal to again r.
02:54
But since it is homomorphism so we can write this part as phi of 1 by q plus phi of 1 by q so on up to phi of 1 by q and this is actually q times so and this is equal to r.
03:19
So that means q into phi of 1 by q is equal to r.
03:25
So phi of 1 by q will be equal to r of r divided by q.
03:31
So this is also satisfying that form.
03:34
Now if we take any general or rational number p by q so y of p by q this can be written as 5 of 1 by q plus so on up to 1 by q and this is written p times p times since it is homomorphism so we can write it as y of 1 by q 5 of 1 by q and so on up to p times or we can write it it as p into phi of 1 by q.
04:13
Now phi of 1 by q we know is equal to r by q.
04:18
So this can be written as r into p by q...