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Technically D1 and D2 can be defined using an alternate definition. However, this notation is rarely used and we will see that it will be unnecessary to consider these two groups. ☐ Example: We will look at D3 , the symmetries of an equilateral triangle. We can rotate by 0 radians, rotate by 2π/3 radians, or rotate by 4π/3 radians and the result is a triangle with the same orientation. If we rotate by 6π/3 = 2π radians, this is equivalent to no rotation at all. We note that rotation by 2π/3 twice is the same as rotation by 4π/3.

10. Prove that for the finite group G with identity e and order 2k for k ∈ N , there is an element g = e such that g · g = e 11. An element of the group G is idempotent if g · g = g . Prove that every group has one and only one idempotent element. 3 Cyclic Groups We have already seen that the complex n th roots of unity Cn form a group under complex multiplication. If we plot all of these points in the complex plane, we see that they are all located on a unit circle. Adjacent points are separated by an angle of 2π/n .

Com 42 An Introduction to Abstract Algebra Group Theory It has been noted that associativity has been inherited from Z+ . Since a + (b + c) = (a + b) + c in Z+ , when we work modulo n , this is also true. [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.

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