Prove that every group of order 45 has a normal subgroup of

Prove that every group of order 45 has a normal subgroup of order 9.

Solution

A non-empty set A is said to form a group, if it satisfies the following propertiess

1. a, b € A Implies a*b € A ( Closure property)

2. a*(b*c) = (a*b)*c (Associative Property)

3. a*b=e = b*a ( Inverse Property)

4. a*e = a = e*a ( Identity Property)

A group is called normal sub-group, if ng^-1 € A Implies gn^-1€ A. A group of order 45 = order (9) * order(5) = gn^-1. So every group group of order 45 has a normal sub-group of order 9 or of order5.