# Thread: Groups of Order 8

1. ## Groups of Order 8

Prove that no pair of the following groups of order 8: I8 (congruence class mod 8), I4 x I2, I2 x I2 x I2, D8 (dihedral group), Q are isomorphic.

The link http://www.mathhelpforum.com/math-he...tml#post392182 shows how D8 and Q are not isomorphic, but I have no idea how to prove the rest of the pairs are not isomorphic.

2. Originally Posted by johnt4335
Prove that no pair of the following groups of order 8: I8 (congruence class mod 8), I4 x I2, I2 x I2 x I2, D8 (dihedral group), Q are isomorphic.

The link http://www.mathhelpforum.com/math-he...tml#post392182 shows how D8 and Q are not isomorphic, but I have no idea how to prove the rest of the pairs are not isomorphic.

Well, the first 3 are abelian: $\mathbb{Z}_8\,,\,\mathbb{Z}_4\times \mathbb{Z}_2\,,\,\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2$ , so they can't be isomorphic with the non-abelian quaternion and dihedral groups.
Now, how these three aren't isomorphic between them? Easy: count number of elements. For example, $\mathbb{Z}_8$ has an element of order 8 and none of the other groups has such an element, and all the elements in the third group have order 2, but not all in the second one...

Tonio