Klein four-group

In mathematics,Klein four-group (or just Klein group or Vierergruppe, often symbolized byletter V), named after Felix Klein, isgroupfour elements,smallest non-cyclic group.

Its multiplication tablegiven by:

	1	a	b	c
1	1	a	b	c
a	a	1	c	b
b	b	c	1	a
c	c	b	a	1

It may be visualized assymmetry group ofrectangle:

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the four elements being:identity,vertical reflection,horizontal reflection, and180 degree rotation.

All elements ofKlein group (exceptidentity) have order 2. Itabelian,is isomorphicC₂ × C₂,direct producttwo copies ofcyclic grouporder 2. Italso isomorphic todihedral grouporder 4.

The essential symmetry betweenthree elementsorder 2the Klein four-group can be seen by its permutation representation on 4 points:

V = < (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) >

In this representation, V isnormal subgroup ofalternating group A₄ (and alsosymmetric group S₄) on 4 letters. AccordingGalois theory,existence ofKlein four-group (andparticular, this particular representation) explainsexistence offormulacalculatingroots of quartic equationstermsradicals.

One can also think ofKlein four-group asautomorphism group offollowing graph:

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