Algebra group structures for Klein four-group
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 . 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:
Verification of properties
- 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.
- 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 is the Klein four-group: It's easy to see that the multiplication coincides precisely with the addition, so the additive structure of 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:
Algebra structure over the field of four elements
The algebra is one-dimensional, and the multiplication is identically zero.