Notes on Algebraic Structures,Group, Examples on Group | Grade 12 > Mathematics > Elementary Group Theory | KULLABS.COM

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Algebraic Structure

A set with one or more binary operations gives rise to what is commonly known as an algebraic structure. In particular, the set Z of integers under the addition ‘+’ is an algebraic structure. It is commonly denoted by (Z, +). In the same way, the set of rational numbers Q under the usual multiplication operation ‘x’, and denoted by (Q, x), is another algebraic structure. A more complicated algebraic structure is a set of real numbers R, together with the usual operations: addition + and multiplication ‘x’. Such an algebraic structure is denoted by (R, +, x). Algebraic structures with one or more binary operations are given special names depending upon additional properties involved.

An algebraic structure consisting of a set G under the operation * on G and denoted by (G,*),may enjoy one or more of the following characteristics:

Given a, b, c, … as the elements of set G, the algebraic structure (G,*) may be

• $$Closed\;if\;a\;*\;b\in\;G\;for\;each\;a,\;b\;\in\;G.$$
• $$Commutative\;if\;a\;*\;b=b*a\in\;G\;for\;each\;a,\;b\;\in\;G.$$
• $$Associative\;if\;(a\;*\;b)*c=a*(b*c)\in\;G\;for\;each\;a,\;b,\;c\;\in\;G.$$
• Existence of identity element. : There exists an element e such that a*e = a = e * a where e is called the identity element.
• The existence of inverse element : There exists an element a’ such that a * a’ = e = a’ * a where a is called the inverse element.

Group

An algebraic structure (G, *), where G is a non-empty set with an operation ‘*’defined on it, is said to be a group, if the operation ‘*’ satisfies the following axioms (called group axioms).

• $$(G1)\;Closure\;axiom\;:\;G\;is\;closed\;under\;the\;operation\;i.e.\;a\;*\;b\in\;G\;for\;each\;a,\;b\;\in\;G.$$
• $$(G2)\;Associative\;Axiom\;:\;The\;binary\;operation\;*\;is\;associative.\;i.e.\;(a\;*\;b)*c=a*(b*c)\in\;G\;for\;each\;a,\;b,\;c\;\in\;G.$$
• $$(G3)\;Identity\;Axiom\;:\;There\;exists\;an\;element\;e\;\in\;G\;such\;that\;a*e\;=\;a\;=\;e\;*\;a\;where\;e\;is\;called\;the\;identity\;element\;of\;a\;with\;respect\;to\;’*’\;in\;G.\;$$
• $$(G4)\;Inverse\;Axiom\;:\;Each\;element\;of\;G\;posseses\;inverse\;i.e.\;for\;each\;element\;\in\;G,\;there\;belongs\;an\;element\;b\;\in\;G\;such\;that\;a*b=e=b*a$$

Thus, an algebraic structure with a set G under a binary operation *, and denoted by (G,*), is known as a group if it is associative, has an identity and an inverse element.

(Note : In the definition of a group sometimes the closure property is also mentioned. Once we mention operation * as a binary operation this becomes redundant.)

Finite and infinite group

A group may contain a finite or an infinite number of elements. It is said to be finite or infinite according as the number of elements is finite or infinite. The number of elements in group is often called order of the group. It is denoted by |G| or (G).

The set S = {1,-1} is a finite group under multiplication; and its order is 2. But, the set of integers is an infinite group under addition.

Trivial group

A group consisting of only one element is called trivial group. The set G= {0} with the usual addition operation is a trivial group, and the set H={1} with the usual multiplication is a trivial group under multiplication.

Abelian group

A group (G,*) is said to be an abelian group if

$$a*b=b*a\;for\;all\;a,\;b\in\;G.$$

Example 1: Show that the set Z of integers does not form a group under the operation defined as

$$x*y=x-y\;for\;every\;x,\;y\;\in\;Z\;$$

Solution:

Here, Z={…, -3,-2,-1,0,1,2,3, …}

Since, the difference x-y of any two integers is also an integer so belongs to Z, hence operation is closed.

For associativity:

(2*3)*4=(2-3)*4=(-1)*4=-1-4=-5

And 2*(3*4) = 2*(3-4)=2*(-1)=2-(-1)=3

$$2*(3*4)\neq\;(2*3)*4$$

i.e. associativity is not satisfied.

Hence Z does not form a group under the given operation.

Example 2: Let G=Q – {1}, the set of all the rational numbers without the unit number. Suppose an operation * is defined on G is given by (a*b) = a + b – ab . Show that a system is a group.

Solution:

• Here, (a*b) = a + b - ab is obviously a rational number Q. It cannot be 1.

If (a*b) = a + b – ab = 1, then (a-1) (b-1) = 0

Or, a = 1 or both 1, which is not possible.

So, * is closed; and the operation is binary.

• $$For\;a,\;b,\;c\;\in\;G,\;(a*b)*c=(a+b-ab)*c$$

$$=a+b-ab+c-(a+b-ab)*c$$

$$=a+b-ab+c-ac-bc+abc$$

$$And\;a*(b*c)=a*(b+c-bc)=a+b+c-bc-a(b+c-bc)$$

$$=a+b-ab+c-ac-bc+abc$$

$$Hence,\;(a*b)*c=a*(b*c),\;(associativity)\;$$

• $$Since\;(a*0)=a+0-0.a=a\;\therefore\;0\;is\;an\;identity\;element.\;$$
• Suppose b is the inverse of a. Then (a*b)=0, the identity element.

Hence,

$$(a*b)=a+b-ab=0$$

$$or,\;b=\frac{a}{a-1}\;\in\;G,\;because\;a\;\neq\;1\;$$

$$i.e.,\;the\;inverse\;of\;a\;is\;\frac{a}{a-1}.$$

• Hence, from the above axioms, G is a group
• Taken reference from

( Basic mathematics Grade XII and A foundation of Mathematics Volume II and Wikipedia.com )

• A set with one or more binary operations gives rise to what is commonly known as an algebraic structure. In particular, the set Z of integers under the addition ‘+’ is an algebraic structure
• An algebraic structure (G, *), where G is a non-empty set with an operation ‘*’defined on it, is said to be a group if the operation ‘*’ satisfies the following axioms (called group axioms).
• Finite and infinite group

A group may contain a finite or an infinite number of elements. It is said to be finite or infinite according as the number of elements is finite or infinite. The number of elements in group is often called order of the group. It is denoted by |G| or (G).

The set S = {1,-1} is a finite group under multiplication; and its order is 2. But, the set of integers is an infinite group under addition.

Trivial group

A group consisting of only one element is called trivial group. The set G= {0} with the usual addition operation is a trivial group, and the set H={1} with the usual multiplication is a trivial group under multiplication.

Abelian group

A group (G,*) is said to be an abelian group if

$$a*b=b*a\;for\;all\;a,\;b\in\;G.$$

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