Answer Happy

Problem 1.5. Show that every abelian group of order 8 is isomorphic to exactly one group in the list Z3, Z2 x Z4, Z8 Classifying non-abelian groups of order 8 Recall we know of two non-abelian groups of order 8: = Dihedral group D4 = {e, a, a, a, b, ba, ba?, ba?} with product determined by the relations at = e, 62 = e and bak = a-kb. This last relation is equivalent to the relation bab-1 = al. Quaternion Group Q is the set of matrices in GL(2,C) 0 n = 0 0 i ni = 1 = (62) i= (• -) j=(-15) k= (;o) = nj 0 ( C 0 nk = i 0 Computation shows that = i’ = j = 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 D4 D4 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 EQ of order 4, every element of Q () is of order 4.