- #1

binbagsss

- 1,265

- 11

I have the following:

Let ##\Omega ## be a discrete subgroup of ##C##, the complex plane.

If:

i) ##\Omega = \{nw_1 | n \in Z\} ##, then ##\Omega ## is isomorphic to ##Z##.

ii) ##\Omega = \{nw_1 + mw_2 | m,n \in Z\} ## where ##w_1/w_2 \notin R ## , then ##\Omega## is isomorphic to ##Z## x ##Z##

So from what I understand isomorphic is a map that is one to one between two sets that preserves the binary relatione exisising between elements, that is ##f(x*y)=f(x)*f(y)## (1), where ##*## is the operation the map is isomorphic to. So to define a isomorphism you need to define:

- two sets

- the map between them

- the relevant operation which is preserved, defined by (1)

QUESTION 1)

So, my book doens't say which operation, is it addition, it also doesn't say which map - is the map to take the integer with the map ##f = n ## in case i) and ##f=n+m## in case 2, under the operation addition it is then easy to show that (1) is obeyed in both cases?

QUESTION 2)

By the wording it seems to imply the fact that ##w_1/w_2 \notin R ## is significant for there to be an isomorphism to ##Z## x## Z##, I don't at all understand why, can someone explain?

Many thanks in advance