Prove Zs, Gs (the group of symmetries of the square) and the quaternion group Q are not pairwise isomorphic.

Question

Prove $Zs, Gs$ (the group of symmetries of the square) and the quaternion group $Q$ are not pairwise isomorphic.

How would you go about proving. Seems quite difficult. I know that none of the latter two can be isomorphic to $\mathbb{Z}_8$ because it is abelian and they are not....

Answer 1

Hint:

• The group of symmetries of the square has two elements of order 4.
• The quaternion group has six elements of order 4.

Answer 2

(I'm assuming that by Zs you mean the $\mathbb Z$ mod 8 under addition, ie the cyclic group of order 8.)

Consider the order of the elements in each group, ie the number of times you can add/multiply each element by itself until you get the identity back. Each of these orders will divide the order of the group, but won't necessarily be equal to the order of the group.