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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

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Group theory

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  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

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