In this representation, V is a normal subgroup of the alternating group A 4 and also the symmetric group S 4 on four letters. In fact, it is the kernel of a surjective group homomorphism from S 4 to S 3. In the construction of finite rings , eight of the eleven rings with four elements have the Klein four-group as their additive substructure. In a similar fashion, the group of units of the split-complex number ring , when divided by its identity component, also results in the Klein four-group.

The simplest simple connected graph that admits the Klein four-group as its automorphism group is the diamond graph shown below. It is also the automorphism group of some other graphs that are simpler in the sense of having fewer entities. These include the graph with four vertices and one edge, which remains simple but loses connectivity, and the graph with two vertices connected to each other by two edges, which remains connected but loses simplicity.

In music composition the four-group is the basic group of permutations in the twelve-tone technique. In that instance the Cayley table is written; [2]. From Wikipedia, the free encyclopedia. This article is about the mathematical concept. Basic notions.

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## Klein four-group

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## Gil' : Norm estimates for resolvents of linear operators in a Banach space and spectral variations

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