Group Theory - Revision history

Iwiseman at 05:10, 18 April 2026

2026-04-18T05:10:46Z

← Older revision		Revision as of 05:10, 18 April 2026
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	=Introduction=		=Introduction=
	A group is a ~~'''~~set~~'''~~ of ~~'''~~elements~~'''~~ equipped with a binary operator that satisfies the following properties.		A group is a set of elements equipped with a binary operator that satisfies the following properties:
	* Closure		* Closure
	* Identity Element		* Identity Element
	* Every element has an inverse		* Every element has an inverse
	* Associativity		* Associativity

	=Binary Operator=		=Binary Operator=
	A binary operator is ~~a type of~~ operator that takes ~~tow~~ operands to perform an operation (or calculation) e.g. + - / * %		A binary operator is an operator that takes two operands to perform an operation (or calculation), e.g. + - / * %.
	~~e.g~~ a . b
	a and b are the ~~'''~~operands~~'''<br>~~		Example: a . b
	. is the ~~'''~~operator~~'''<br>~~		a and b are the operands
	~~So we have~~		"." is the operator
	*Arithmetic Operators + -
	*Comparison Operators < > = !=		Examples of binary operators:
	*Boolean Operators AND, OR, XOR~~, NOT~~		* Arithmetic Operators: + -
			* Comparison Operators: < > = !=
			* Boolean Operators: AND, OR, XOR

	=Closure=		=Closure=
	The operation must be closed. When the operator is applied to any two elements in the set the result must also be ~~and~~ element in the set. ~~So this can work on~~ infinite ~~and~~ finite ~~sets given~~		The operation must be closed.
	ℤ₅ = {0,1,2,3,4}		When the operator is applied to any two elements in the set, the result must also be an element in the set.

	Operation = addition mod 5		Closure works for both finite and infinite sets.
	Test closure
			Example using a finite set:
	3 + 4 = 7 → 7 mod 5 = 2 (in the set)		ℤ₅ = {0,1,2,3,4}
	4 + 4 = 8 → 8 mod 5 = 3 (in the set)
	1 + 3 = 4 (in the set)		Operation = addition mod 5
			Test closure:

			3 + 4 = 7 → 7 mod 5 = 2 (in the set)
			4 + 4 = 8 → 8 mod 5 = 3 (in the set)
			1 + 3 = 4 (in the set)

	=Identity Element=		=Identity Element=
	*The identity element I must be unique		* The identity element I must be unique.
	*There cannot be more than one		* There cannot be more than one.
	*There must exist an ~~identity~~ element I in the set such that for every element in a ~~in the set~~		* There must exist an element I in the set such that for every element a:
	a . I = I . a = a
	~~For example~~ Integers under ~~multiplication I~~ = ~~1 where~~ a = 5		a . I = I . a = a
	a * 1 = 1 * a ~~= 1~~ = a
	~~5 * 1 = 1 * 5 = 1 = 5~~		Examples:
	~~The identity element I must be unique~~		* Integers under addition → identity is 0
	~~There cannot be more than one~~		a + 0 = 0 + a = a

			* Integers under multiplication → identity is 1
			a * 1 = 1 * a = a

	=Inverse=		=Inverse=
	*Every element in the set must have an inverse		* Every element in the set must have an inverse.
	*For each element a in the set, there must exist an element a₍ᵢₙᵥ₎ such that:		* For each element a in the set, there must exist an element a₍ᵢₙᵥ₎ such that:
	a . a₍ᵢₙᵥ₎ = a₍ᵢₙᵥ₎ . a = I
	~~For example: Integers~~ under addition ~~where a =5~~		a . a₍ᵢₙᵥ₎ = a₍ᵢₙᵥ₎ . a = I
	a + (-a) = (-a) + a = 0
	5 + (-5) = (-5) + 5 = 0		Example (integers under addition):
			a + (-a) = (-a) + a = 0
			5 + (-5) = (-5) + 5 = 0

	=Associativity=		=Associativity=
	Associativity ensures that the result of applying the operation does ~~'''~~not~~'''~~ depend on how the elements are ~~parenthesitised~~. ~~<br>~~		Associativity ensures that the result of applying the operation does not depend on how the elements are parenthesised.
	~~For example~~
	(a.b) . c = a. (b.c)		(a . b) . c = a . (b . c)

Iwiseman: /* Inverse */

2026-04-18T05:06:30Z

Inverse

← Older revision		Revision as of 05:06, 18 April 2026
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	*Every element in the set must have an inverse		*Every element in the set must have an inverse
	*For each element a in the set, there must exist an element a₍ᵢₙᵥ₎ such that:		*For each element a in the set, there must exist an element a₍ᵢₙᵥ₎ such that:
	a * a₍ᵢₙᵥ₎ ~~mod n~~ = a₁		a . a₍ᵢₙᵥ₎ = a₍ᵢₙᵥ₎ . a = I
			For example: Integers under addition where a =5
			a + (-a) = (-a) + a = 0
			5 + (-5) = (-5) + 5 = 0
			=Associativity=
			Associativity ensures that the result of applying the operation does '''not''' depend on how the elements are parenthesitised. <br>
			For example
			(a.b) . c = a. (b.c)

Iwiseman: /* Inverse */

2026-04-18T05:00:09Z

Inverse

← Older revision		Revision as of 05:00, 18 April 2026
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	=Inverse=		=Inverse=
	*Every element in the set must have an inverse		*Every element in the set must have an inverse
	*For each element a in the set, there must exist an element ~~ainv~~ such that:		*For each element a in the set, there must exist an element a₍ᵢₙᵥ₎ such that:
	a * ~~a_inv~~ mod n = ~~a_1~~		a * a₍ᵢₙᵥ₎ mod n = a₁

Iwiseman at 04:57, 18 April 2026

2026-04-18T04:57:27Z

← Older revision		Revision as of 04:57, 18 April 2026
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	The identity element I must be unique		The identity element I must be unique
	There cannot be more than one		There cannot be more than one
			=Inverse=
			*Every element in the set must have an inverse
			*For each element a in the set, there must exist an element ainv such that:
			a * a_inv mod n = a_1

Iwiseman at 04:48, 18 April 2026

2026-04-18T04:48:31Z

← Older revision		Revision as of 04:48, 18 April 2026
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	4 + 4 = 8 → 8 mod 5 = 3 (in the set)		4 + 4 = 8 → 8 mod 5 = 3 (in the set)
	1 + 3 = 4 (in the set)		1 + 3 = 4 (in the set)
			=Identity Element=
			*The identity element I must be unique
			*There cannot be more than one
			*There must exist an identity element I in the set such that for every element in a in the set
			a . I = I . a = a
			For example Integers under multiplication I = 1 where a = 5
			a * 1 = 1 * a = 1 = a
			5 * 1 = 1 * 5 = 1 = 5
			The identity element I must be unique
			There cannot be more than one

Iwiseman at 04:32, 18 April 2026

2026-04-18T04:32:44Z

← Older revision		Revision as of 04:32, 18 April 2026
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	a and b are the '''operands'''<br>		a and b are the '''operands'''<br>
	. is the '''operator'''<br>		. is the '''operator'''<br>
			So we have
			*Arithmetic Operators + -
			*Comparison Operators < > = !=
			*Boolean Operators AND, OR, XOR, NOT
			=Closure=
			The operation must be closed. When the operator is applied to any two elements in the set the result must also be and element in the set. So this can work on infinite and finite sets given
			ℤ₅ = {0,1,2,3,4}

			Operation = addition mod 5
			Test closure

			3 + 4 = 7 → 7 mod 5 = 2 (in the set)
			4 + 4 = 8 → 8 mod 5 = 3 (in the set)
			1 + 3 = 4 (in the set)

Iwiseman: Created page with "=Introduction= A group is a '''set''' of '''elements''' equipped with a binary operator that satisfies the following properties. * Closure * Identity Element * Every element has an inverse * Associativity =Binary Operator= A binary operator is a type of operator that takes tow operands to perform an operation (or calculation) e.g. + - / * % e.g a . b a and b are the '''operands'''
. is the '''operator'''
"

2026-04-18T04:20:01Z

Created page with "=Introduction= A group is a '''set''' of '''elements''' equipped with a binary operator that satisfies the following properties. * Closure * Identity Element * Every element has an inverse * Associativity =Binary Operator= A binary operator is a type of operator that takes tow operands to perform an operation (or calculation) e.g. + - / * % e.g a . b a and b are the '''operands'''<br> . is the '''operator'''<br>"

New page

=Introduction=
A group is a '''set''' of '''elements''' equipped with a binary operator that satisfies the following properties.
* Closure
* Identity Element
* Every element has an inverse
* Associativity
=Binary Operator=
A binary operator is a type of operator that takes tow operands to perform an operation (or calculation) e.g. + - / * %
e.g a . b
a and b are the '''operands'''<br>
. is the '''operator'''<br>