I.N.Herstein in Topics of Algebra defines a group as a set having a special element $i$ such that:
$a,i\in A(S) $ which satisfies $i\cdot a = a\cdot i = a$
In this way, this follows commutativity holds true for identity element, then doesnt it run contradictory to the non-necessary condition that $a \cdot b \neq b \cdot a$ ?
In similar vein is the set of integers under subtraction a group? {my reasoning is no it isnt, because $a-0 = a \neq 0-a$}
Soham