Lets call group G. Suppose , abelian group is when ab=ba.
How to prove that G is abelian if and only if
Ok, lets start from and hopefully we will get into abelian group definition.
, , so . Now since G is group, for all elements, except identity element, should exist inverse: ,.
Lets multiply by from left, and from right.
, , and this is criteria of abelian group.
Proving backward is exactly the same, first we say , and multiply both sides by a from left and b from right – , , point proven.