Algebra group structures for cyclic group:Z4

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This article gives specific information, namely, algebra group structures, about a particular group, namely: cyclic group:Z4.
View algebra group structures for particular groups | View other specific information about cyclic group:Z4

There exists a unique way of making cyclic group:Z4 into an algebra group over field:F2. It is not an algebra group over any other field. This is unique in a strict sense, not merely up to isomorphisms.

Summary

Algebra Smallest n for which it can be embedded in UT(n,2) Maximum of element nilpotencies Nilpotency index of whole algebra
the only possible one, where one generator squares to the other 3 3 3

Algebra structure

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 b 0
b 0 0

Verification of properties

  • N is associative: All products of length three 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 three 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 cyclic of order 4: Indeed, (1 + a)^2 = 1 + b and (1 + b)^2 = 1, so 1 + a is an element of order four (see also powering map by field characteristic is same in algebra and 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 & 1 \\ 0 & 0 & 0 \\\end{pmatrix}, \qquad b =  \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\\end{pmatrix}