Algebra group structures for 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.