Cyclic groups
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Cyclic Groups
A cyclic group is a group generated by powers of an element a in the group. In order to understand this definition we must first explain what we mean by the power of an element. As a concrete example let us consider the group [Diagram goes here - download the original to see it.] [Diagram goes here - download the original to see it.] Powers of the element 1 are defined by Diagram goes here - download the original to see it.] In terms of the group structure (G, 0) nth power of the element [Diagram goes here - download the original to see it.] is the elements. [Equation goes here - download the original to see it.] where the composition is taken n times This example demonstrates that ¢6 is some power of 1. [Diagram goes here - download the original to see it.] To the definition of the power of an element [equation], we add [Equation goes here - download the original to see it.] , the identity element of G [Equation goes here - download the original to see it.] is the inverse of element a. The set of all powers generated by an element [Equation goes here - download the original to see it.] is denoted by [equation]. It is the set [Equation goes here - download the original to see it.] Thus [Equation goes here - download the original to see it.] The powers of each element of a group G generates a cyclic subgroup of G. For example, for ¢6, consider the powers of 2. [Equation goes here - download the original to see it.] Thus Equation goes here - download the original to see it.] Likewise [Equation goes here - download the original to see it.] Thus [Equation goes here - download the original to see it.] Thus [Equation goes here - download the original to see it.] are distinct, proper cyclic subgroups of the group ¢6. As the above examples indicate the symbol [symbol] is often used to signify the cyclic group of order n. An alternative notation of the same cyclic group is [Equation goes here - download the original to see it.]. The order of a group is the number of elements in the group. The order of G is designated by the symbol ½G½ For example ½¢6½ = 6 The order of an element a of a group G is the number of elements in the cyclic subgroup [Equation goes here - download the original to see it.]. [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] So if G is a cyclic group of order s, then it follows from the definition of a cyclic group that there exists some element [Equation goes here - download the original to see it.] - the element a generates the set S under the group operation. A group is cyclic if there exists at least one element that generates the whole group, whose order is equal to the order of the group. With these definitions and observations, further properties of cyclic groups can be explored, as in the following example. Example [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Solution [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.]
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Contents of
Cyclic groups
1 Cyclic Groups
2 The Cayley-Hamilton theorem
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