Why Read This Book? - Index of

248 Chapter 8 Groups
Example 8.1.17 Here is another example of a finite abelian group. In
Exercise 2.2.5, you observed that x 2 =−1 has no real solution x. Nothing prevents
us from creating a symbol, say i, declaring i 2 =−1, and then noting that i is
not a real number. Now consider the set S ={±1, ±i} with binary operation ×
defined according to Table 8.3. Then (S, ×, 1, −1 ) is an abelian group.
× 1 −1 i −i
1 1 −1 i −i
−1 −1 1 −i i
i i −i −1 1
−i −i i 1 −1
(8.3)
Example 8.1.18 Similar to Example 8.1.17, define × on Q ={±1, ±i, ±j, ±k}
according to Table 8.4. The gist of this algebraic structure can be understood
by noticing that i 2 = j 2 = k 2 =−1, but i, j, and k do not commute with each
other. If you think of the letters i, j, and k being written on the face of a clock
at 12, 4, and 8 o’clock, respectively, then multiplication of two different elements
in the clockwise direction produces the third. Multiplication of two elements in
the counterclockwise direction produces the negative of the third. For example,
j × k = i and i × k =−j. These three square roots of −1 are called quaternions,
and they motivate a group with eight elements.
× 1 −1 i −i j −j k −k
1 1 −1 i −i j −j k −k
−1 −1 1 −i i −j j −k k
i i −i −1 1 k −k −j j
−i −i i 1 −1 −k k j −j
j j −j −k k −1 1 i −i
−j −j j k −k 1 −1 −i i
k k −k j −j −i i −1 1
−k −k k −j j i −i 1 −1
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(8.4)
EXERCISE 8.1.19 On the set S ={0, 1, 2, 3, 4, 5}, let ◦ be defined by Table 8.5.
(a) Explain how you know that ◦ is a binary operation.
(b) Is the operation commutative?
(c) What is the identity element?
(d) Determine the inverse of each element.
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