1 / 16

Group theory

Group theory. Group Definition. A group is a set G = {E,  } where E is a set of elements and  is a binary operation on E. For a group we have the following axioms:. Closed under binary operation Asso c iative binary operation Identity element Inverse element. A_001 A_002 A_003

ydowell
Download Presentation

Group theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Group theory

  2. GroupDefinition A group is a set G = {E, } where E is a set of elements and  is a binary operation on E. For a group we have the following axioms: Closed under binary operation Associative binary operation Identity element Inverse element A_001 A_002 A_003 A_004

  3. Identity element Uniqueness T_001 A group have only one identity element Proof:

  4. Inverse elementUniqueness T_002 An element has only one inverse Proof:

  5. Invers element(a-1)-1 = a T_003 The inverse of the inverse of an element is the element itself (a-1)-1 = a Proof:

  6. Identity elementIts own inverse T_004 The identity element is its own inverse e-1 = e Proof:

  7. Inverse of a product T_005 The inverse of a product is the product of the inverse in reverse order (ab)-1 = b-1a-1 Proof:

  8. Inverse of a product T_005 The inverse of a product is the product of the inverse in reverse order (ab)-1 = b-1a-1 Proof:

  9. Summing up A_001 A_002 A_003 A_004 T_001 T_002 T_003 T_004 T_005

  10. SubgroupDef D_002: A subgroup H is a subset of a group G that itself is a group with the same binary operation as G. For a subgroup we must have: H subset Closed under binary operation Identity element Inverse element

  11. SubgroupTheorem T_006: A subset H is a subgroup if and only if ab-1 H for all a,b H. Proof:

  12. GroupExample - Number G E a b ab e a-1 Undergruppe av

  13. y l1 l2 D C GroupExample - Rotation x A B D A C B s1 speiling om x-aksen r0 rotasjon 00 A D B C C D C B s2 speiling om y-aksen r1 rotasjon 900 B A D A B B C A s3 speiling om diagonalen l1 r2 rotasjon 1800 A D C D D A A D r3 rotasjon 2700 s4 speiling om diagonalen l2 C B B C

  14. y l1 l2 D C GroupExample - Rotation x A B r0 rotasjon 00 s1 speiling om x-aksen r1 rotasjon 900 s2 speiling om y-aksen r2 rotasjon 1800 s3 speiling om diagonalen l1 s4 speiling om diagonalen l2 r3 rotasjon 2700 D A A D D C s2 r1-1 = s2 = C B B C A B s2 r1-1 = s4 D A D C = s4 C B A B

  15. y l1 l2 D C GroupExample - Rotation x A B r0 rotasjon 00 s1 speiling om x-aksen r1 rotasjon 900 s2 speiling om y-aksen r2 rotasjon 1800 s3 speiling om diagonalen l1 s4 speiling om diagonalen l2 r3 rotasjon 2700

  16. END

More Related