Definition of Groups

A group G is a finite or infinite set of elements together with a binary operation (called the group operation) that together satisfy the four fundamental properties of closure, associativity, the identity property, and the inverse property.

G1(closure): for all X and Y in G

X*Y is in G

G2(Associativity):For all X,Y and Z in G

(X*Y)*z=X*(Y*Z).

G3(Identity). There is an element e in G such that

e*X=X*e=X

for all X in G.

G4(inverse).for all X in G there is an X’ in G such that

X*X’=X’*X=e

Examples of Groups

1.The integers modulo n, often denoted by Zn, form a group under addition.

2.Group of integers Z with addition as operation and zero as identity element.

3 . The rational numbers form a group under addition.

4 . The positive rationals form a group under multiplication with identity element 1.

5 . The positive rationals form a group under multiplication with identity element 1

6 . The real numbers form a group under addition.

7 . The complex numbers form a group under addition.

8 . The set of all non-zero real numbers is group under the operation of multiplication.

9 . The set of all non-zero complex numbers is group under the operation of multiplication.

10. The set of all 2×2 matrices

[a b]

[c d]

with real (or complex) entries satisfying ad−bc 6= 0 is a group under the operation of matrix multiplication

11 . The set of all 2×2 matrices

[a b]

[c d]

with real (or complex) entries satisfying ad−bc = 1 is a group under the operation of matrix multiplication.

12 . The set of all non-zero rational numbers is group under the operation of multiplication