Suppose H, K C G are subgroups of orders 5 and 8, respectively. Prove that H K = {e}.

Step-by-step explanation:

Consider the provided information.

We have given that H and k are the subgroups of orders 5 and 8, respectively.

We need to prove that H∩K = {e}.

As we know "Order of element divides order of group"

Here, the order of each element of H must divide 5 and every group has 1 identity element of order 1.

1 and 5 are the possible order of 5 order subgroup.

For subgroup order 8: The possible orders are 1, 2, 4 and 8.

Now we want to find the intersection of these two subgroups.

Clearly both subgroup H and k has only identity element in common.

Thus, H∩K = {e}.