A monoid is an algebraic structure consisting of a set together with an associative binary operation and an identity element. A group is a special kind of monoid that also has an inverse element for each element in the set. In other words, a monoid is called a group if it satisfies the additional requirement of having inverses.

Let's break down the given options and understand why option D is the correct answer:

Option A: (a*a) = a = (a c)b
This option does not define a group. It suggests that the binary operation (denoted by *) is commutative, which means that a * b = b * a. However, a group does not necessarily require the binary operation to be commutative.

Option B: (a*c) = (a c)c
This option does not define a group either. It suggests that the binary operation is associative, which is a property of a monoid. However, it does not mention anything about the existence of inverses, which is a requirement for a group.

Option C: (a c) = a*d
This option does not define a group. It suggests that the binary operation is not associative since it is not clear how (a c) and a*d are related. A group requires the binary operation to be associative.

Option D: (a*c) = (c*a) = e
This option defines a group. It states that for every element a in the set, there exists an element c such that a * c = c * a = e, where e is the identity element. This condition ensures that every element has an inverse, which is a requirement for a group.

Conclusion:
Among the given options, option D is the correct answer because it defines a monoid as a group by stating the existence of inverses for every element in the set. The other options either do not define a group or do not satisfy the necessary conditions for a group.