Why Read This Book? - Index of

246 Chapter 8 Groups
Example 8.1.9 Consider S ={0, 1, 2, 3, 4, 5} and let ⊕ be described as in
Table 8.1.
⊕ 0 1 2 3 4 5
0 0 1 2 3 4 5
1 1 2 3 4 5 0
2 2 3 4 5 0 1
3 3 4 5 0 1 2
4 4 5 0 1 2 3
5 5 0 1 2 3 4
(8.1)
That ⊕ is well defined is immediate, for there is a unique value in each position
in the table. Closure of ⊕ is also obvious, since every entry in the table is an
element of S. �
EXERCISE 8.1.10 Is ⊕ is commutative? How can you tell? Is there an identity
element? Does every element have an inverse?
One way to verify associativity of ⊕ would be to do all possible calculations
of the form (a ⊕ b) ⊕ c and a ⊕ (b ⊕ c) to check if they are equal. In Section 8.3,
we will construct this algebraic structure formally and prove associativity as a
theorem.
Example 8.1.11 Let S be the set in Example 8.1.9 and define the operation ⊗
as in Table 8.2.
⊗ 0 1 2 3 4 5
0 0 0 0 0 0 0
1 0 1 2 3 4 5
2 0 2 4 0 2 4
3 0 3 0 3 0 3
4 0 4 2 0 4 2
5 0 5 4 3 2 1
(8.2)
EXERCISE 8.1.12 Is ⊗ commutative? Is there an identity element? Does every
element have an inverse?
8.1.2 Groups Defined
We have names to refer to algebraic structures with certain sets of features. Here
is our first such name.
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