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Homework answers / question archive / Problem 2
¯e, i, j, k satisfying (¯e)^2 = e, i^2 = j^2 = k^2 = ijk = ¯e is called the quaternion group Q8.
a) Prove that ¯e ∈ Z(Q8). (Hint: Show that ¯e commutes with ¯e, i, j and k)
b) It is customary to denote the identity of Q8 by 1 (instead of e) and ¯e by −1. For a ∈ Q8, define −a = (−1)a = a(−1). Prove that Q8 = {1, −1, i, −i, j, −j, k, −k}, and that |Q8| = 8. Is Q8 abelian? Provide justification for your answer. (Hint: To prove |Q8| = 8 you must show that ±1, ±i, ±j, ±k are distinct)
c) Show that Q8 does not contain a pair of non-trivial complementary subgroups. Using the below problem is helpful to conclude that Q8 is not isomorphic to D4.
(below problem):
Let n ≥ 3 be an integer. Consider the dihedral group Dn. Prove that Dn ∼= H |x K for subgroups H and K of Dn with K <|/ Dn and |K| = n.
|x is the semi direct product of H and K
<|/ is normal subgroup i.e, K is a normal subgroup of Dn.
