Question

Define Klein- 4 group? ​

Answer 1

Step-by-step explanation:

In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one. It can be described as the symmetry group of a non-square rectangle (with the three non-identity elements being horizontal and vertical reflection and 180-degree rotation), as the group of bitwise exclusive or operations on two-bit binary values, or more abstractly as Z2 × Z2, the direct product of two copies of the cyclic group of order 2. It was named Vierergruppe (meaning four-group) by Felix Klein in 1884. It is also called the Klein group, and is often symbolized by the letter

Answer 2

Note :

Group : An algebraic system (G,*) is said to be a group if the following condition are satisfied :

1. G is closed under *
2. G is associative under *
3. G has a unique identity element
4. Every element of G has a unique inverse in G

Moreover , if a group (G,*) also holds commutative property , then it is called commutative group or abelian group .

Cyclic group : A group G is called a cyclic group , if there exists an element a ∈ G , such that every element x ∈ G can be written as x = aⁿ for some integer n . And the element a is called the generator of G .

Solution :

Please refer to the attachments .