In mathematics , the Klein group or the Vierergruppe (in German, group of four ) is the abelian group of four elements in which each element is the inverse of itself. It receives this name in honor of the German Félix Klein .
Let C 2 = {1,2} be the cyclic group of order two; that is: 2 2 = 1. Also let H = C 2 xC 2 = {(1, 1), (1, 2), (2, 1), (2, 2)}. Then the product of ordered pairs in C 2 xC 2 can be performed . For example (1, 2)) x (2, 1) = (1×2, 2×1) = (2,2). Other cases: (2, 2) x (2, 2) = (1,1) [1] It is observed that the product of any element by itself is the unit = (1,1), so there is no generator in H. So H is not isomorphic to the cyclic group C 4 = {1, b, b 2 , b 3 } of order 4. We also have (1; 1) 2 = (1; 1), (1, 2) 2 = (1,1), (2,1) 2= (1,1) and (2; 2) 2 = (1,1), the order of each element is 2, except for the identity = (1,1), just as each element is inverse of itself.
- With the identification 1 = (1,2); a = (1,2); b = (2,1) and c = (2,2) the table that appears in the following section is generated.
Summary
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- 1 Group definition
- 2 Algebraic properties
- 3 See also
- 4 References and notes
Group definition
The Klein group is the group (K, •) where K = {1, a, b, c} and whose internal binary operation • is defined by the following table:
| • | one | to | b | c |
| one | one | to | b | c |
| to | to | one | c | b |
| b | b | c | one | to |
| c | c | b | to | one |
Sometimes, since c = a • b , we write K = {1, a, b, ab} . With these names, the operation table is
| • | one | to | b | ab |
| one | one | to | b | ab |
| to | to | one | ab | b |
| b | b | ab | one | to |
| ab | ab | b | to | one |
Algebraic properties
- The Klein group is isomorphic to the direct productof the cyclic group of order 2 by itself: K = Z 2 x Z 2 .
- Klein’s group is abelian [2], that is, the internal operation is commutative.
- The Klein group is not cyclical, so there is no generating element and, therefore, it is not isomorphic to the Z 2 group .
- The order of all elements is 2, except for the neutral element 1 whose order is 1. [3]
- Klein’s group has the representation <a, b | a 2= b 2 = (ab) 2 = 1> [4]
- Klein’s group only has three of its own subgroups, isomorphic to the cyclic group of order 2. They are generated by each of the elements other than neutral 1.
- The group of rhombus turns is isomorphic to the Klein group [5]
- There is an alternating subgroup of order 4, from the symmetric group S 4that is isomorphic with the Klein group.
