Group 6 ?: r: b' m9 QA group is defined as a finite or infinite set of Operands& W! x$ R! Y) { w7 q* b
(called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator - Q5 Q: L) e, H: e: M to form well-defined products and which furthermore satisfy the following conditions: + [2 w" Z& v( J" b8 s; J$ M1. Closure: If and are two elements in , then the product is also in . ! u) t; d5 M$ J4 p, ^2. Associativity: The defined multiplication is associative, i.e., for all , . 8 D# G/ }2 q' S" v0 i' ^8 R# p3. Identity: There is an Identity Element5 y! Y/ M) V0 ~6 r0 B
(a.k.a. , , or ) such that for every element . + Y2 R; I ]5 v) C, G6 r# F p- j
4. Inverse: There must be an inverse or reciprocal of each element. Therefore, the set must contain an element such that for each element of . 5 S; y9 |4 J0 ]' `$ m4 c2 lA group is therefore a Monoid& d2 P0 V6 U7 Y, U
for which every element is invertible. A group must contain at least one element. - w# e3 w9 K" y u! k4 Z" J! r5 VThe study of groups is known as Group Theory 6 y5 p, f$ M G Q' B. If there are a finite number of elements, the group is called a Finite Group# J7 }- m. |5 d! V1 C! E [4 Y
and the number of elements is called the Order2 a6 C2 B( d. ]8 p$ Y: ^
of the group. 8 N4 D9 U5 I. t9 a$ J* W
& m" m( \7 S% D8 Z$ F. b5 ^
Since each element , , , ..., , and is a member of the group, group property 1 requires that the product ; Q8 L0 s6 R# d! ^; [4 h, z
& t5 z1 m( D+ ?: F& ^; Q( o9 D" b
(1) 5 C* a* q" W. L; g
6 G2 Q+ P$ s2 U; U" y8 _
! A2 i ^7 N$ O# ~* P* a
; H7 `. {6 M! B2 F Q# J$ H- i
must also be a member. Now apply to , 2 i* m3 |- J0 k. ?/ a* G; G# U
' x; A! p- I7 I* r& f& K
+ N0 ^- H1 }4 _% c! b4 a' T
(2) # N3 n# f7 {) d$ i& z+ ^6 F p$ B
0 i, e4 }# [3 x. a% h 9 v; K, |' W0 Y1 Z# @( Z* l' E; L) t # S; T- z P2 k0 L% [+ S% ]But $ t* J/ B' v4 e1 B0 F& u
; f7 q1 z+ x/ r$ Z+ Q7 ^
% o& g) W$ u, C1 ]$ q9 t6 [" M
, b, U. {" N' P9 z
: U$ B C$ R' V7 S4 Z
6 T0 s# e! F1 h/ U9 G( _
(3)2 w8 l( c* U9 E0 R6 X( o& _
so , U1 d% V* o" @ z8 I
6 X0 t3 z3 l4 k' i: l
(4) " g! k6 U, K- \& q
0 N! A! K: `. R5 Y% R' X _3 ^) x. n& ^2 l
. P+ G ?4 }0 [* [
which means that - L5 D `3 V Y
7 I: m2 c' @! H# S/ P/ h
(5) & g3 o* z' E6 \! D0 V2 Z5 E
2 |9 W& u) \& d0 g; y. ^
' H/ r, j! l$ s0 _0 H* X5 M
+ g$ e6 q" Q! w* X* A) Q
and 3 C: m- p' k: N7 z7 ]5 }$ x
2 Y: l4 |- U: z1 _
(6) ( I: d& j' M0 {
! \0 j& Z2 }! Q9 @
S, H9 k# d5 {8 e
$ v3 D/ M9 C N9 a) z) C & W; j6 W, u9 n0 \$ \ 7 L( D9 }! k3 {0 \: p