Let Q be the group of rational numbers and Z the group of in

Let Q be the group of rational numbers and Z the group of integers, where Z x Q is the group of ordered pairs (x,y) with x in Z and y in Q under component-wise addition. Let M* be a subset of Z x Q such that M* = {(x,y) : y = m*z for some z in Z, m in Q}. Show that M* is a normal subgroup of Z x Q. Please be concise, and validate any claims. Thank you.

Solution

Consider (a,b) + (x,y)

where (x,y) is any element in M*

and a,b any element in Z X Q

(a,b) + (x,y) = (a+x , b+y) = (x+a ,y+b) = (x,y) + (a,b)

b is element in Q

y = m*z , m in Q , z in Z

b + y = r some element in Q