Discussion on definition of group Well, I am little bit confuse in definition of group. My question is this that why we need of identity property in definition of group because if a set has a associative binary operation such that every element has inverse then it obviously contain identity element. If I'm wrong then can someone give me an example of a set which not group just because of identity property i.e. we have a set equipped with an associative binary operation along with inverse property but it does not have identity element. Please clear my doubt.
 A: Indeed it would be possible to define a group as a (nonempty) semigroup equipped with a unary function $\cdot\,^{-1}$ that satisfies
$$ a(a^{-1})b = ba(a^{-1}) = (a^{-1})ab = b(a^{-1})a = b $$
for all $a$ and $b$. There are four identities because we need inverses and identities to both work from the left and right.
In fact, if we want to be even more minimalist we can get down to two identities:
$$ a(a^{-1})b = b = b(a^{-1})a $$
together with associativity. This is enough to make sure that $ax=b$ and $xa=b$ always have solutions, which by this question implies that our structure is a group.
Why is it usually not done this way?
There's no real technical pressure to make the number of symbols in the signature or axiom system as small as possible. As long as everyone agrees what are and are not groups, what really matters is not how tersely you can state an axiomatic definition, but how easily you can present the definition and its usual elementary consequences in an introductory textbook -- and for that, being too parsimonious is not necessarily a win.
Additionally, doing it this way around would obscure the fact that groups are monoids that satisfy additional conditions. It can be proved, of course, but it's more instructive to have it hold immediately by definition.
Especially because beginning students of group theory are often not very mathematically sophisticated (it is often the first completely abstract definition of an algebraic structure one sees), everything that makes it harder to work with the definition is generally to be avoided.
A: It depends what you mean by the "inverse property". You might be after the theory of inverse semigroups.
A semigroup is a set under an associative binary operation, while an inverse semigroup $S$ is a semigroup in which for every element $x \in S$ there exists a unique $y\in S$ such that $x = xyx$ and $y = yxy$ - here, $x$ and $y$ are inverse elements. See Wikipedia or these notes of Mark Lawson for more details.
Groups are inverse semigroups, but the converse is not true. For example, $\{0, 1\}$ under multiplication is an inverse semigroup which is not a group (both elements are self-inverse, so it is an inverse semigroup, but there is a $0$ so it is not a group). Another example is the following multiplication table from Wikipedia, which again defines an inverse semigroup which is not a group (multiplication is not bijective).
\begin{array}{c|ccccc}
&a & b & c & d & e\\\hline
a& a& a& a& a& a\\
b& a& b& c& a& a\\
c& a& a& a& b& c\\
d& a& d& e& a& a\\
e& a& a& a& d& e
\end{array}
More examples can be found in this old Math.SE question.
An important class of examples of inverse semigroups which are not groups are the Symmetric inverse semigroups $\mathcal{I}_X$ for each set $X$; these are the "partial" bijections (that is, bijections which are not defined everywhere) on $X$ under a certain natural composition operation. These play a similar role in inverse semigroup theory as the symmetric groups play in group theory, as every inverse semigroup can be embedded into an symmetric inverse semigroup (this is the Wagner-Preston theorem).
A: Here is what is usually done.
Whenever you encounter a binary operation, ask the following three questions:

*

*Is the binary operation commutative?


*Is the binary operation associative?


*Is there an identity element?
Now, in case the answer to the third question is "yes", you ask the fourth question.


*What about the inverse?

Notice that it would be completely meaningless to talk about the inverse if there is no identity.
