Algebra group structures for Klein four-group

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This article gives specific information, namely, algebra group structures, about a particular group, namely: Klein four-group.
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There exists a unique way of making cyclic group:Z4 into an algebra group over field:F2. There is also a unqiue way of making the group an algebra group over field:F4, which is compatible with the former.

Algebra structure over the field of two elements

Multiplication table (structure constants)

The algebra is two-dimensional. It has the following multiplication table, with basis elements a,b. Note that Adjoint group of a radical ring is abelian iff the radical ring is commutative‎, so the algebra group is commutative, and we don't have to worry about the order of multiplication:

a b
a 0 0
b 0 0

Verification of properties

  • N is associative: All products of length two or more involving basis elements are zero, hence, by linearity, all products of length three or more are zero.
  • N is nilpotent: All products of length two or more involving basis elements are zero, hence, by linearity, all products of length three or more are zero.
  • The algebra group for N is the Klein four-group: It's easy to see that the multiplication coincides precisely with the addition, so the additive structure of N is the same as the multiplicative structure of its algebra group.

Description as subalgebra of niltriangular matrix Lie algebra

The algebra can be realized explicitly as a subalgebra of niltriangular matrix Lie algebra:NT(3,2) as follows:

a = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\\end{pmatrix}, \qquad b =  \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\\end{pmatrix}

Algebra structure over the field of four elements

The algebra is one-dimensional, and the multiplication is identically zero.