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For instance u · v = r1 and v · u = r2 . · e r1 r2 v u d e e r1 r2 v u d r1 r1 r2 e u d v r2 r2 e r1 d v u v v d u e r2 r1 u u v d r1 e r2 d d u v r2 r1 e To summarize, D3 is a group of order 6 and is a nonabelian group. ☐ Notice that the dihedral group can be nonabelian. Although there are some elements that commute, such as r1 and r2 , there are pairs such as u, v that do not. Dihedral groups are important because they are one of the more straightforward examples of nonabelian groups. They provide an example that is easy to think about and even manipulate by hand.

0] is the identity element. For inverses [0] is its own inverse and [1] and [2] are inverses of each other. Although the type of addition seen in this example may seem unnatural, we use it every day when we look at a clock. Just as 5 + 10 = 3 mod 12 , five hours after 10 o’clock is 3 o’clock. This type of example will be a major one that we will continually return to throughout the book. ☐ We see from the above examples that the group structure is very much dependent on both the set we are using as well as the binary operation on the set.

In this case the inverse of the element a is denoted by −a . 2 Examples of Groups To see how far ranging the definition of a group is, we will look at an extensive series of examples of groups. Some of what follows constitute counterexamples. These are examples of sets with binary operation that fail to meet some part of the definition of a group. Example: x1. ∈ Z under addition is an abelian group. Addition is associative, 0 is an identity, and for every x ∈ Z we have −x as an inverse. Since x ∈ Z satisfies these axioms, it forms a group under addition.

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