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AUTOMORPHISMS With Examples.pptx
- 2. What is/are Automorphism/s? β’ An automorphism is an isomorphism from a group to itself. β’ The set of all automorphisms of G forms a group, called the automorphism group of G, and denoted Aut(G). An automorphism is simply a bijective homomorphism of an object with itself. An automorphism is an endomorphism (i.e., a morphism from an object to itself) which is also an isomorphism (in the categorical sense of the word, meaning there exists a right and left inverse endomorphism).
- 3. One of the earliest group automorphisms (automorphism of a group, not simply a group of automorphisms of points) was given by the Irish mathematician William Rowan Hamilton in 1856, in his icosian calculus, where he discovered an order two automorphism, writing: so that π is a new fifth root of unity, connected with the former fifth root π by relations of perfect reciprocity. Source: Wikipedia
- 4. β’ Homomorphism οΌpreserve the group operation οπ π₯ β π¦ = π π₯ β π(π¦) οπ π₯ + π¦ = π π₯ + π(π¦) β’ Isomorphism οΌone-to-one (1-1) οΌOnto- a function such that every element in B is mapped by some element A οΌHomomorphism Automorphis m A map of Ο: πΊ β πΊ is said to be automorphism if Ο is 1-1, onto, and homomorphism.
- 5. EXAMPLE (β, +) π: β β β π π + ππ = π β ππ π π§1 = π π§2 β π π1 + π1π = π π2 β π2π β π1 + π1π = π2 β π2π β π1 = π2 πππ π1 = π2 β π1 + π1π = π2 + π2π β π§1 = π§2 β΄ π ππ πππ β ππ β πππ π§ = π + ππ β β π§β² = π β ππ β β β π π§β² = π(π β ππ) = π + ππ = z β΄ π ππ ππππ π π§1 + π§2 = π π1 + π2 + π π1 + π2 = π1 + π2 β π(π1 + π2) = π1 β π1π + π2 β π2π = π π1 + π1π + π π2 + π2π = π π§1) + π(π§2 β΄ π ππ ππππππππππππ πππ β ππ β πππ ππππ ππππππππππππ AUTOMORPHISM
- 6. πΌ β π΄π’π‘ β€10 πΌ: β€10 β β€10 πΌ 1 πΌ π πΌ π = πΌ 1+1+β―+1 π π‘ππππ = πΌ 1 +πΌ 1 +β―+πΌ 1 π π‘ππππ = πΆ π π πΌ 1 =π΄π’π‘(β€10) Theorem 6.2 Properties of Isomorphic Acting on Elements Property 4. πΊ = 1 ππ πππ ππππ¦ ππ πΊ = πΌ(1) β β€10 = 1 β β€10 = πΌ(1) EXAMPLE π¨ππ(β€ππ) Generators of β€10 are relatively prime to it: π, π, π, π πΌ 1 π 1 β πΌ1 3 β πΌ3 7 β πΌ7 9 β πΌ9
- 7. π¨ππ β€ππ = πΌ1, πΌ3, πΌ7, πΌ9 πΌ1 β πππππ‘ππ‘π¦ πΌ3 πΌ3 π₯ = πΌ3 π¦ β 3x(mod 10) = 3y (mod 10) β x(mod 10) = y (mod 10) πΌ3 1 = 3 3 is generator of β€ππ πππ β ππ β πππ ππππ πΌ3 π + π = 3 π + π = πΌ3 π) + πΌ3(π ππππππππππππ 3x mod 10 πΌ3 1 β πΌ3 3 β πΌ3 7 β πΌ3 9 β 3 9 1 7 πΌ3 π + π = πΌ3 π) + πΌ3(π πΌ3 1 + 2 = πΌ3 1) + πΌ3(2 = 3 1) + 3(2 ππ = 3 + 6 πΌ3 3 β 9
- 8. Uses of Automorphisms β’ Automorphisms of groups can be used as a means of constructing new groups from the original group. β’ We can use the automorphisms group machinery to determine the characteristic subgroups of a group. β’ In computer science, automorphisms are useful in understanding the complexity of algebraic problems.