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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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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
Identity element Uniqueness T_001 A group have only one identity element Proof:
Inverse elementUniqueness T_002 An element has only one inverse Proof:
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:
Identity elementIts own inverse T_004 The identity element is its own inverse e-1 = e Proof:
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:
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:
Summing up A_001 A_002 A_003 A_004 T_001 T_002 T_003 T_004 T_005
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
SubgroupTheorem T_006: A subset H is a subgroup if and only if ab-1 H for all a,b H. Proof:
GroupExample - Number G E a b ab e a-1 Undergruppe av
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
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
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