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Unformatted text preview: Cyclic Group Supplement Theorem 1. Let u1D454 be an element of a group u1D43A and write ⟨ u1D454 ⟩ = uni007B.alt02 u1D454 u1D458 : u1D458 ∈ ℤ uni007D.alt02 . Then ⟨ u1D454 ⟩ is a subgroup of u1D43A . Proof. Since 1 = u1D454 , 1 ∈ ⟨ u1D454 ⟩ . Suppose u1D44E , u1D44F ∈ ⟨ u1D454 ⟩ . Then u1D44E = u1D454 u1D458 , u1D44F = u1D454 u1D45A and u1D44Eu1D44F = u1D454 u1D458 u1D454 u1D45A = u1D454 u1D458 + u1D45A . Hence u1D44Eu1D44F ∈ ⟨ u1D454 ⟩ (note that u1D458 + u1D45A ∈ ℤ ). Moreover, u1D44E − 1 = ( u1D454 u1D458 ) − 1 = u1D454 − u1D458 and − u1D458 ∈ ℤ , so that u1D44E − 1 ∈ ⟨ u1D454 ⟩ . Thus, we have checked the three conditions necessary for ⟨ u1D454 ⟩ to be a subgroup of u1D43A . Definition 2. If u1D454 ∈ u1D43A , then the subgroup ⟨ u1D454 ⟩ = { u1D454 u1D458 : u1D458 ∈ ℤ } is called the cyclic subgroup of u1D43A generated by u1D454 , If u1D43A = ⟨ u1D454 ⟩ , then we say that u1D43A is a cyclic group and that u1D454 is a generator of u1D43A . Examples 3. 1. If u1D43A is any group then { 1 } = ⟨ 1 ⟩ is a cyclic subgroup of u1D43A . 2. The group u1D43A = { 1 , − 1 , u1D456, − u1D456 } ⊆ ℂ ∗ (the group operation is multiplication of complex num bers) is cyclic with generator u1D456 . In fact ⟨ u1D456 ⟩ = { u1D456 = 1 ,u1D456 1 = u1D456,u1D456 2 = − 1 ,u1D456 3 = − u1D456 } = u1D43A . Note that − u1D456 is also a generator for u1D43A since ⟨− u1D456 ⟩ = { ( − u1D456 ) = 1 , ( − u1D456 ) 1 = − u1D456, ( − u1D456 ) 2 = − 1 , ( − u1D456 ) 3 = u1D456 } = u1D43A . Thus a cyclic group may have more than one generator. However, not all elements of u1D43A need be generators. For example ⟨− 1 ⟩ = { 1 , − 1 } ∕ = u1D43A so − 1 is not a generator of u1D43A . 3. The group u1D43A = ℤ ∗ 7 = the group of units of the ring ℤ 7 is a cyclic group with generator 3. Indeed, ⟨ 3 ⟩ = { 1 = 3 , 3 = 3 1 , 2 = 3 2 , 6 = 3 3 , 4 = 3 4 , 5 = 3 5 } = u1D43A. Note that 5 is also a generator of u1D43A , but that ⟨ 2 ⟩ = { 1 , 2 , 4 } ∕ = u1D43A so that 2 is not a generator of u1D43A . 4. u1D43A = ⟨ u1D70B ⟩ = { u1D70B u1D458 : u1D458 ∈ ℤ } is a cyclic subgroup of ℝ ∗ . 5. The group u1D43A = ℤ ∗ 8 is not cyclic. Indeed, since ℤ ∗ 8 = { 1 , 3 , 5 , 7 } and ⟨ 1 ⟩ = { 1 } , ⟨ 3 ⟩ = { 1 , 3 } , ⟨ 5 ⟩ = { 1 , 5 } , ⟨ 7 ⟩ = { 1 , 7 } , it follows that ℤ ∗ 8 ∕ = ⟨ u1D44E ⟩ for any u1D44E ∈ ℤ ∗ 8 . If a group u1D43A is written additively, then the identity element is denoted 0, the inverse of u1D44E ∈ u1D43A is denoted − u1D44E , and the powers of u1D44E become u1D45Bu1D44E in additive notation. Thus, with this notation, the cyclic subgroup of u1D43A generated by u1D44E is ⟨ u1D44E ⟩ = { u1D45Bu1D44E : u1D45B ∈ ℤ } , consisting of all the multiples of u1D44E . Among groups that are normally written additively, the following are two examples of cyclic groups....
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This document was uploaded on 12/28/2011.
 Fall '09

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