Answer Happy

Recall we know of two non-abelian groups of order 8: Dihedral group De = {e, a, a’, a, b, ba, baa, ba} with product determined by the relations a4 = e, 62 = e and bak = a-kb. This last relation is equivalent to the relation bab-1 = a?. Quaternion Group Q is the set of matrices in GL(2,C) 1 n = 의 0 0 0 = ni = 0 0 = (69) i= (6-5) j= (-10) 0:0) nj GE CO k= nk = i 0 Computation shows that n i? = ja = k” = n ij = k, jk = i, ki =j ji = nk, kj = ni, ik = nj = We see that these two groups are non-isomorphic since: Case DA DA has an element a of order 4 with some (in fact every) element of D4 – (a) of order 2. Case Q For every element x e Q of order 4, every element of Q - (2) is of order 4. Problem 1.6. In this problem you are going to classify the non-abelian groups of order 8 = 1. Show that every non-abelian group of order 8 has an element of order 4. 2. Let G be any group of order 8. Show that if x, y E G are of order 4 then x2 = y2. 3. Let G be a non-abelian group of order 8. We know we must have an element of order 4. Now we have two possible cases