Explain why 3 and 4 are not isomorphic and D8 and 8 are not

Explain why:
/3 and /4 are not isomorphic
and
D₈ and /8 are not isomorphic

Solution

a) All three groups have order 24. But Z/24Z and Z/3Z × Z/8Z have elements of order 24 (1 and (1, 1) respectively, for example); thus both are cyclic and isomorphic to each other. By contrast, 12 times any element of Z/4Z × Z/6Z is the identity; thus it isn’t cyclic and can’t be isomorphic to either of the other two.

b) Now Q8 has only one element of order 2 and therefore cannot contain any subgroups isomorphic

to Z/2Z × Z/2Z. Also, Z/8Z is cyclic and hence does not contain a subgroup isomorphic to the

group Z/2Z × Z/2Z because that group is not cyclic.