Groups

    Group is a non-empty set ogether with a binary operation,*,defined on G ,which satisfies the following axioms.
G satisfies:
                1)Closure property
               2)Associative
               3)Identity
               4)Inverse
             with respect to binary operation *,if(Z4,+4) is a group.
      Examples of groups:
               1)The set of all symmetries of the infinite ornamental pattern in which the arrow heads are                    spaced,uniformly,a unit part along a line is an abelian group,under composition.Let T                          denote a translation to the right by one unit.,the inverse of t,a translation to the left by one                    unit and H a reflection across the horizontal line of the figure.Then every member of the                       group is of the form x1,x2,...xn.where each xi belongs to {T,T^-1,H}.
              2)Let F be any of Q,R,C or Zp(p is a prime).The set GL(2,F) of all 2x2 matrices with non-                      zero determinants and entries from F is a non-Abelian group under matrix multipication.
              3)For a fixed point (a,b) in R^2 ,define T[a,b]:R^2->R^2 by (x,y)->(x+a,y+b).Then
                 T(R^2)={T[a,b]|a,b belongs to R } is a group under function composition.
              4)The set R^n={(a1,a2,....an)|a1,a2....an belongs to R} is a group under component wise                           addition[ie,(a1,a2,...an)+(b1,b2,...bn)=(a1+b1,a2+b2,....an+bn)].
              5)For all integers n>=1 ,the set of complex roots of                                                unity{cos(k.360'/n)+isin(k.360'/n)|k=0,1,2,...n-1}
               6)Klein group
                  It has 4 elements
                     [1,0,0,1],[-1,0,0,-1],[-1,0,0,1],[1,0,0,-1]
                     The binary operation is matrix multiplicatio.This is the smallest group which is not                               cyclic.
               7)Paul's Group:(Quatterian group)
                  There are 8 elements:+1,-1,+1,-1,+j,-j,+R,-R
                   where 1=[1,0,0,1]
                               i=[i,0,0,-i]
                              j=[0,1,-1,0]
                              R=[0,i,i,0]
                                where i is the complex i=(0,1)
                  It is a group under matrix addition and multiplication
                  NB:The elements of the matrix are given in the order [a11,a12,a21,a22]
           8)The set R* of non-zero real numbers is a group under ordinary multiplication.The identity is                1.The inverse of a is 1/a.
           9)SLn(R)={Anxn:A is a nxn matrix with entries from R and det(A)=1}
               Then (SLn(R),*) where * is the matrix multiplication.
               SLn is called the special linear group.
           10)Un (C)={Anxn :A is an nxn matrix such that A*A =I}
                (Un(c),X)  where X is a matrix multiplication,then this is a group.It is called unitary group. where A,* is the complex conj of A A*=[aji^_]if A=[aij]
            11)Set Q+ of positive rationals is a groupunder ordinary multiplication.
            12){1,2,3,4} under multiplication modulo 5
         13)all 3x3 matrices with real entries of the form                               [a11=1,a12=a,a13=b,a21=0,a22=1,a23=c,a31=0,a32=0,a33=1]
           14)G ,a matrix under multiplication operatio,whereG={[a11=a,a12=a,a13=a,a14=a]|a belongs to R,a!=0}