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Klein Four Group

The Klein four-group typeset structure (after the name Vierergruppe introduced by Felix Klein for it in his Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade in 1884) is the (multiplicative) group typeset structure, the direct product of two copies of the cyclic group of order 2.  

It can be defined on the set typeset structure endowed with the multiplication given by typeset structure and typeset structure.  Its Cayley table is  

1 i j k 1 1 i j k i i 1 k j j j k 1 i k k i j 1

Another description typeset structure is that is it the multiplicative group of reduces residues modulo 8:

_ _ _ _ 1 3 5 7 _ _ _ _ _ 1 1 3 5 7 _ _ _ _ _ 3 3 1 7 5 _ _ _ _ _ 5 5 7 1 3 _ _ _ _ _ 7 7 5 3 1

It is smallest non-cyclic group, and it is Abelian.

Klein four group is the symmetry group of a rhombus (or of a rectangle, or of a planar ellipse), with the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180 degree rotation.

It is also the automorphism group of the graph with four vertices and two disjoint edges.

It can be also defined through permutations on a 4-element set typeset structure: typeset structure(34), typeset structure, and typeset structure.

Since the generators typeset structure can be represented as the product of two even permutations , typeset structure is a normal subgroup of the symmetric group typeset structure ., and consequently also of the alternating group typeset structure .

typeset structure has three genuine subgroups or order 2: typeset structure, typeset structure, and typeset structure. We also have the composition series
typeset structure, typeset structure.
The factors of the series are cyclic of order 2,3,2,2.

Klein four group is actually the dihedral group typeset structure.

Cite this web-page as:

Štefan Porubský: Klein Four Group.

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