Some properties of group theory

f is a group, then ( ) .
Proof: By definition, ( ) is inverse to and a is also the inverse
of , then ( ) , since by theorem the inverse is unique.
Theorem 1.17: If is a group, then ( )
Proof: It suffices to show that
( ) ( )
( ) ( )
A similar computation leads to ( ) . Thus, we get the result.
Theorem 1.18: Every element in a group is cancelable.
Proof: Consider the equation since the inverse of a is exist, then
( ) ( ) ( ) ( ) .
Theorem 1.19: If a is an element in a group G, then
1) .
2) ( ) , .
Lemma 1.20: If a and b are noncommuting elements of a group G, then
the elements of a set * + are all distinct.
Theorem 1.21: Any noncommutative group has at least six elements.
Definition 1.22: Let n be a fixed positive integer. Two integers a and b
are said to be congruent modulo n, written ( ), if and only if
Theorem 1.23: Let n be a fixed positive integer and a and b are arbitrary
integers. Then ( ) have the same nonnegative remainder
when divided by n.
Theorem 1.24 : Let n be a fixed positive integer and a,b, c and d are
arbitrary integers. Then
1) ( ).
2) If ( ) then ( ).
3) If ( ) and ( ), then ( ).