DEFINITIONS AND EXAMPLES OF GROUPS 13 1.8 Let G be a group in which every nonidentity element is an involution. Show that G is abelian. NOTE: An abelian group in which every nonidentity element has the same prime order p is called an elementary abelian p-group. 1.9 Consider the eight objects ±1, ±i, ±j and ±k with multiplication rules: ij = k jk = i ki = j ji = —f c kj = —i ik = j where the minus signs behave as expected and 1 and 1 multiply as expected. (For example, (—1)7 = —j and (—/)(—j) = ij = k.) Show that these objects form a group containing exactly one involution. NOTE: This is called the quaternion group and is denoted g8-
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