Question

How will you differentiate between c and d groups in group theory?

Answer 1

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Answer 2

Let’s begin with the more general definition concerning sets .

A Set (mathematics) is essentially a collection of any sort of objects .These objects are called the elements or the members of the set . Here are some more details :

A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is generally also ignored (unlike a list or multiset). Members of a set are often referred to as elements and the notation a∈Aa∈A is used to denote that aa is an element of a set AA . The study of sets and their properties is the object of set theory. Older words for set include aggregate and set class.

Source : Set -- from Wolfram MathWorld

From a historical point of view , the study of sets originally started with Georg Cantor as a way to investigate the theory of infinite series . Set theory has played an important role in the foundations of mathematics .

Now Let’s see what is a group.

A Group (mathematics) is an algebraic structure and a set closed under certain specific operations . There is an identity element in the set related to an associative binary operation , and every element has an inverse within the set .

For more details :

A group GG is a finite or infinite set of elements together with a binary operation (called the group operation) that together satisfy the four fundamental properties of closure, associativity, the identity property, and the inverse property. The operation with respect to which a group is defined is often called the "group operation," and a set is said to be a group "under" this operation. Elements A,B,C,…A,B,C,… with binary operation between AAand BB denoted ABAB form a group if

1. Closure: If AA and BB are two elements in GG , then the product ABAB is also in GG .

2. Associativity: The defined multiplication is associative, i.e., for all A,B,C∈GA,B,C∈G , (AB)C=A(BC)(AB)C=A(BC).

3. Identity: There is an identity element II(a.k.a. 11 ,EE , or ee ) such that IA=AI=AIA=AI=Afor every element A∈GA∈G .

4. Inverse: There must be an inverse (a.k.a. reciprocal) of each element. Therefore, for each element AA of GG, the set contains an element B=A−1B=A−1 such that AA−1=A−1A=IAA−1=A−1A=I.

A group is a monoid each of whose elements is invertible.

A group must contain at least one element, with the unique (up to isomorphism) single-element group known as the trivial group.

The study of groups is known as group theory. If there are a finite number of elements, the group is called a finite groupand the number of elements is called the group order of the group. A subset of a group that is closed under the group operation and the inverse operation is called a subgroup. Subgroups are also groups, and many commonly encountered groups are in fact special subgroups of some more general larger group.



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