Let H be a subgroup of the group Z with H Z Suppose that 617

Let H be a subgroup of the group Z with H Z. Suppose that 617 H. Can H contain 2016 or 304?. Justify your answers, of course.

Note :Please answer this question depend on Z is agroup not cyclic group at subject of abstract algebra 2nd edition for charles C. Pinter

I got wrong answered from chegg before so I submit again to get correct answer

Solution

Z is a group of all positive and negative integers, including 0. Thus Z includes both 2016 and 304 as these two numbers are integers. H is a subgroup such that H Z. Thus all the integers are not in H. However, we do not know which integers are in H and which are not. If H = {0, 1, 617, 304, 2016 }, then both 2016 and 304 H and still H Z. If H = {0, 1,2,3, 304, 617} , then H Z and 304 H but 2016 is not in H. If H = {0,1, 2, 617, 2016}, then H Z , and 2016 H , but 304 is not in H. Further, if H = {0,1, 2, 3, 617} , then H Z and neither of 304 and 2016 is in H. Thus, all the possibilities exist.