Show that Show that The Additive groups Ropf and Qopf are no

Show that:

Show that: The Additive groups Ropf and Qopf are not isomorphic

Solution

By cantor\'s diagonal argument, there is no possible bijection between Q and R. Since an isomorphism needs to be a bijection, there is no possible isomorphism between the additive groups R and Q

.Recall that R is a group with respect to addition of real numbers and that
Q = R {0} is a group with respect to multuplication of real numbers. The
identity element in R is 0 belongs to R and the identity element in Q is 1 BELONGS TO R.
First, notice that every nontrivial (that is, nonzero) element in R has infinite
order. Indeed, if a 2 R, a 6= 0 then for n 1 the n-th additive power of a is na
and na 6= 0 for every integer n 1; hence |a| = 1 in R for any a 6= 0. Of course,
the order of 0 in R is equal to 1. In particular, this shows that the group R has no
elements of order 2.
On the other hand, in Q we do have an element of order 2, namely 1. Indeed,
1 = (1)1 6= 1 but (1)2 = 1, so that indeed 1 has order 2 in Q.
Since Q has an element of order 2 but R has no elements of order 2, these groups
are not isomorphic.