Group theory Group Definition A group is a
![Group theory Group theory](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-1.jpg)
![Group Definition A group is a set G = {E, } where E is Group Definition A group is a set G = {E, } where E is](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-2.jpg)
![Identity element Uniqueness T_001 A group have only one identity element Proof: Identity element Uniqueness T_001 A group have only one identity element Proof:](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-3.jpg)
![Inverse element Uniqueness T_002 An element has only one inverse Proof: Inverse element Uniqueness T_002 An element has only one inverse Proof:](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-4.jpg)
![Invers element (a-1)-1 = a T_003 The inverse of the inverse of an element Invers element (a-1)-1 = a T_003 The inverse of the inverse of an element](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-5.jpg)
![Identity element Its own inverse T_004 The identity element is its own inverse e-1 Identity element Its own inverse T_004 The identity element is its own inverse e-1](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-6.jpg)
![Inverse of a product T_005 The inverse of a product is the product of Inverse of a product T_005 The inverse of a product is the product of](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-7.jpg)
![Inverse of a product T_005 The inverse of a product is the product of Inverse of a product T_005 The inverse of a product is the product of](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-8.jpg)
![Summing up A_001 A_002 A_003 A_004 T_001 T_002 T_003 T_004 T_005 Summing up A_001 A_002 A_003 A_004 T_001 T_002 T_003 T_004 T_005](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-9.jpg)
![Subgroup Def D_002: A subgroup H is a subset of a group G that Subgroup Def D_002: A subgroup H is a subset of a group G that](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-10.jpg)
![Subgroup Theorem T_006: A subset H is a subgroup if and only if ab-1 Subgroup Theorem T_006: A subset H is a subgroup if and only if ab-1](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-11.jpg)
![Group Example - Number G E a b ab e a -1 Undergruppe av Group Example - Number G E a b ab e a -1 Undergruppe av](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-12.jpg)
![y l 2 D C Group x Example - Rotation D C l 1 y l 2 D C Group x Example - Rotation D C l 1](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-13.jpg)
![y l 2 D C Group x Example - Rotation A r 0 rotasjon y l 2 D C Group x Example - Rotation A r 0 rotasjon](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-14.jpg)
![y l 2 D C Group Example - Rotation l 1 x A B y l 2 D C Group Example - Rotation l 1 x A B](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-15.jpg)
![END END](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-16.jpg)
- Slides: 16
![Group theory Group theory](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-1.jpg)
Group theory
![Group Definition A group is a set G E where E is Group Definition A group is a set G = {E, } where E is](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-2.jpg)
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: A_001 Closed under binary operation A_002 Associative binary operation A_003 Identity element A_004 Inverse element
![Identity element Uniqueness T001 A group have only one identity element Proof Identity element Uniqueness T_001 A group have only one identity element Proof:](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-3.jpg)
Identity element Uniqueness T_001 A group have only one identity element Proof:
![Inverse element Uniqueness T002 An element has only one inverse Proof Inverse element Uniqueness T_002 An element has only one inverse Proof:](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-4.jpg)
Inverse element Uniqueness T_002 An element has only one inverse Proof:
![Invers element a11 a T003 The inverse of the inverse of an element Invers element (a-1)-1 = a T_003 The inverse of the inverse of an element](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-5.jpg)
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 element Its own inverse T004 The identity element is its own inverse e1 Identity element Its own inverse T_004 The identity element is its own inverse e-1](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-6.jpg)
Identity element Its own inverse T_004 The identity element is its own inverse e-1 = e Proof:
![Inverse of a product T005 The inverse of a product is the product of Inverse of a product T_005 The inverse of a product is the product of](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-7.jpg)
Inverse of a product T_005 The inverse of a product is the product of the inverse in reverse order (ab)-1 = b-1 a-1 Proof:
![Inverse of a product T005 The inverse of a product is the product of Inverse of a product T_005 The inverse of a product is the product of](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-8.jpg)
Inverse of a product T_005 The inverse of a product is the product of the inverse in reverse order (ab)-1 = b-1 a-1 Proof:
![Summing up A001 A002 A003 A004 T001 T002 T003 T004 T005 Summing up A_001 A_002 A_003 A_004 T_001 T_002 T_003 T_004 T_005](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-9.jpg)
Summing up A_001 A_002 A_003 A_004 T_001 T_002 T_003 T_004 T_005
![Subgroup Def D002 A subgroup H is a subset of a group G that Subgroup Def D_002: A subgroup H is a subset of a group G that](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-10.jpg)
Subgroup Def 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
![Subgroup Theorem T006 A subset H is a subgroup if and only if ab1 Subgroup Theorem T_006: A subset H is a subgroup if and only if ab-1](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-11.jpg)
Subgroup Theorem T_006: A subset H is a subgroup if and only if ab-1 H for all a, b H. Proof:
![Group Example Number G E a b ab e a 1 Undergruppe av Group Example - Number G E a b ab e a -1 Undergruppe av](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-12.jpg)
Group Example - Number G E a b ab e a -1 Undergruppe av
![y l 2 D C Group x Example Rotation D C l 1 y l 2 D C Group x Example - Rotation D C l 1](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-13.jpg)
y l 2 D C Group x Example - Rotation D C l 1 A A B B s 1 speiling om x-aksen r 0 rotasjon 00 A B D C C B C D s 2 speiling om y-aksen r 1 rotasjon 900 D A B A B C s 3 speiling om diagonalen l 1 r 2 rotasjon 1800 C D A D D A r 3 rotasjon 2700 s 4 speiling om diagonalen l 2 B C C B
![y l 2 D C Group x Example Rotation A r 0 rotasjon y l 2 D C Group x Example - Rotation A r 0 rotasjon](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-14.jpg)
y l 2 D C Group x Example - Rotation A r 0 rotasjon 00 s 1 speiling om x-aksen r 1 rotasjon 900 s 2 speiling om y-aksen r 2 rotasjon 1800 s 3 speiling om diagonalen l 1 s 4 speiling om diagonalen l 2 r 3 rotasjon 2700 D A C s 2 r 1 -1 = A B l 1 D s 2 B D A C B = B C s 2 r 1 -1 = s 4 D C D A C B = s 4 A B
![y l 2 D C Group Example Rotation l 1 x A B y l 2 D C Group Example - Rotation l 1 x A B](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-15.jpg)
y l 2 D C Group Example - Rotation l 1 x A B r 0 rotasjon 00 s 1 speiling om x-aksen r 1 rotasjon 900 s 2 speiling om y-aksen r 2 rotasjon 1800 s 3 speiling om diagonalen l 1 r 3 rotasjon 2700 s 4 speiling om diagonalen l 2
![END END](https://slidetodoc.com/presentation_image_h/94cd81fdbe44ed9e6600528485a0901e/image-16.jpg)
END
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