cyclicgroup

cyclicgroup - Cyclic Group Supplement Theorem 1. Let u1D454...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

This document was uploaded on 12/28/2011.

Page1 / 4

cyclicgroup - Cyclic Group Supplement Theorem 1. Let u1D454...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online