GROUPS

Definition 1 (Group)

A group is a binary operation structure (G,*) is a nonempty set and * is a binary operation on G such that the following axioms are satisfied.

G1 : For all a, b, c in G,

(a*b)*c = a*(b*c)

Associativity of *

G2 : There is an element e in G such that for all x in G,

e*x = x*e = x

Identity of *

G3 : Corresponding to each a in G, there ia an element a' in G such that

a*a' = a'*a = e

Inverse a' of a

Note : If any of these axioms is not satisfied, then a binary structure is not a group.



Example 1:

(Z,+) is a group because it satisfies G1, G2, G3. That is,

G1 : for all a, b, c in Z

(a+b)+c = a+(b+c)

G2 : Since 0+a = a+0 = a for all a in Z then, 0 is the identity for (Z,+).

G3 : For each a in Z, exist an element a' = -a element of Z such that a+(-a) = 0 = (-a)+a. Then, -a is the inverse element of a in (Z,+).


Example 2 :

Let * be defined on Q+ by a*b = ab/2
Show that the binary structure (Q+,*) is a group.

Solution :

G1 : (To show associativity of *)

Let a, b, c be element of Q+

LHS:
(a*b)*c
= (ab/2)*c
= (ab/2)c / 2
= (ab)c/4
= a(bc)/4 multiplication is associative on Q+.
= a(bc/2) / 2
= a(bc)/2
= a*(b*c)
= RHS

Since, LHS = RHS then, * is associative on Q+.

G2 : Let a in Q+.
(To find (b = e) element of Q+ such that a*b = b*a = a)

LHS : a*b = ab/2 = a --- b = 2 element of Q+

RHS : b*a = ba/2 = a --- b = 2 element of Q+

Then, b = e = 2 is the identity element of (Q+,*).

G3 : (To find the inverse element of (Q+,*))

LHS : a' = 4/a element of Q+.

RHS : a' = 4/a element of Q+.

Then, b = a' = 4/a is the inverse element of (Q+,*).

Therefore, (Q+,*) is a group.

sources : http://www.youtube.com/user/singingbanana