(a) Show that the relations a^4 = 1, b^2 = a^2 and b^(-1)ab = a^(-1) define a group of order 8. (It is called the quaternion group.)(b) Show that this group is not isomporphic to Î4 (also of order 8).Please see the attachment for more precise text description.We have the definition of quaternion group as follows.We also have anther non-abelian group with order 8. The group is called dehedral group defined as with the following definition.Now I claim that and are not isomorphic.

Actually, we only need to check the number of elements with order 2.

In , it only contains one element with order 2. Since , then we have , then , . Then both and has order 4. We note that . Thus only has one element that has order 2.

In , at least we have two elements and that have order 2.

Therefore, and are not isomorphic.You can visit the link http://www.math.kent.edu/~white/41001/order8sub.pdf to know more detailed structures of the two groups.

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