Appendix II - On Near-Fields
Published online by Cambridge University Press: 05 September 2013
Summary
A finite set F, equipped with two operations + and ·, is called a near-field if:
F is a commutative group under the operation + (the identity element is denoted by 0).
F — {0} is a group under the operation · (the identity element is denoted by 1 and juxtaposition is often used to denote this operation).
The law (a + b)c = ac+bc holds for all a, b, c ∈ F.
In [Z], Zassenhaus classified finite near-fields but here we need only certain elementary results.
If F is a near-field, we set L(F) = F × F*, where F* is the multiplicative group F — {0} acting on F by multiplication on the right. Then F* acts regularly on F — {0}. Conversely, if a group M acts on an abelian group F and M is regular on F#, we can equip F with the structure of a near-field such that L(F) ≅ F × M: it suffices to choose an element 1 in F — {0} and to put (1x)(1y) = 1(xy) for x,y ∈ M.
Let F be a near-field. As F* acts transitively on F — {0}, F is an elementary abelian f-group for some prime number f, which is called the characteristic of F.
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- Character Theory for the Odd Order Theorem , pp. 137 - 138Publisher: Cambridge University PressPrint publication year: 2000