Abstract
This chapter concerns near-fields, particularly finite near-fields. It offers the most complete account of finite near-fields in English and contains some new material on the structure of near-fields.
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Notes
- 1.
I did go to some lengths to try to get hold of the book on near-fields by Wähling [141]. It would probably be accessible through university libraries and does not seem to be commercially available, even second hand.
- 2.
In fact, I experienced something familiar to most mathematicians, obtaining Wolf rather late in the development of this book and then discovering that many results I had laboured over were actually well-known (this does not apply to the results that were wrong!).
- 3.
The term is not universal, which is why I use \(\mathfrak {Z}\) rather than “Z”. Robinson himself uses “Z-group” to denote a quite distinct class of groups.
- 4.
Older authors sometimes restrict the term to groups having cyclic commutator subgroups with cyclic quotients. Q8 does not have this property, but it is metacyclic, in modern parlance.
- 5.
The literature contains references to Zassenhaus’s book on group theory [145] for this proof; but it does not seem to appear in the English edition of that work. It seems to me that this is one of several chance happenings that have rendered finite near-field theory less accessible to non-German-speaking audiences.
- 6.
A version of it does also appear in [35] but, frankly, it puzzles me. In that book, condition (ii), above, is replaced by
$$\displaystyle \begin{aligned} \begin{array}{rcl} (p^\beta - 1) \equiv 0 \ \ (\mathit{modulo}\ \mathit{4}) \ \Rightarrow \delta \not \equiv 0 \ \ (\mathit{modulo}\ \mathit{4}) \end{array} \end{aligned} $$(3.42)(page 237). These authors are not the sort of writers to make simple errors, so all I can presently do is report it.
- 7.
Wolf’s formulation of the lemma is a little different, in that he specifically requires the group to have an element with order 2(a−1). Since we already know this to be the case for N ∗, I miss that out.
- 8.
There is also a group commonly known as the full icosahedral group. It is quite distinct from the binary icosahedral group but does have the same order because it is (A 5 ⊕C2)—how to make a difficult subject harder!
- 9.
Proving this is not remotely straightforward—as is true for many of the “facts” stated here.
- 10.
That is, the normal subgroup generated by the normal nilpotent subgroups. In finite cases, as we have here, this is the unique maximal normal nilpotent subgroup of G.
- 11.
Recall the Jordan-Hölder theorem from group theory.
- 12.
\(\frac {3}{2}\)-transitive permutation groups are transitive groups G acting on sets Ω in such a way that for each α ∈ Ω, G α ≠ {Id}, and the subgroup {G α} acts transitively on { Ω− α}.
- 13.
I think it is also known as the binary dodecahedral group and is a unique stem extension (a special sort of central extension) with normal subgroup C2 and quotient A 5. More recently, it has also been called the icosian group in [29] and other publications of John Horton Conway, whose death, from Covid-19, was announced on the day I wrote this. The Irish mathematician, W. R. Hamilton, invented an Icosian game, which was sold commercially in the 1860s and involved finding a Hamiltonian cycle around the edges of a dodecahedron, represented physically as a peg board.
- 14.
Berliners might describe it as a schlimmbesserung—an improvement that makes things worse!
References
P.J. Cameron, in Permutation Groups. London Mathematical Society Student Texts, vol. 45 (1999)
J.R. Clay, Near Rings: Geneses and Applications (Oxford Science Publications, Oxford, 1992)
J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups (Springer, Berlin, 1999)
P. Dembowski, in Finite Geometries. Classics in Mathematics (Springer, Berlin, 1991)
J.D. Dixon, B. Mortimer, Permutation Groups. Graduate Texts in Mathematics (Springer, Berlin, 1991)
B.J. Gardner, L. Shaoxue, R. Wiegandt, Rings and Radicals (Longman, London, 1994)
D. Gorenstein, Finite Groups (AMS Chelsea Publishing, New York, 1980)
T. Grundhöfer, H. Zassenhaus, A criterion for infinite non-Dickson near-fields of dimension two. Results Math (1989)
T. Grundhöfer, C. Hering, Finite near-fields classified again. J. Group Theory 20, 829–839 (2017)
K.W. Gruenberg, in Cohomological Topics in Group Theory. Lecture Notes in Mathematics, vol. 143 (Springer, Berlin, 1970)
M. Hall, The Theory of Groups, 2nd edn. (AMS Chelsea Publishing, 1999)
I.M. Isaacs, in Finite Group Theory. Graduate Studies in Mathematics, vol. 92 (American Mathematical Society, Providence, 2008)
B. Huppert, N. Blackburn, Finite Groups III (Springer, Berlin, 1982)
G. Karpilovsky, The Jacobson Radical of Classical Rings (Longman, London, 1991)
S. Lang, Algebra (Revised Third Edition) (Springer, Berlin, 2002)
G. Michler, in Theory of Finite Simple Groups. New Mathematical Monographs, vol. 8 (Cambridge, 2006)
C.P. Milies, K.S. Sudarshan, An Introduction to Group Rings (Kluwer, Dordecht, 2002)
G. Mullen, C. Mummert, in Finite Fields and Applications. Student Mathematical Library, vol. 41 (American Mathematical Society, Providence, 2000)
D.S. Passman, Permutation Groups (Dover, New York, 2012)
G. Pilz, Near-Rings: The Theory and Its Application, 2nd edn. (Elsevier, Amsterdam, 1983)
D.J.S. Robinson, A Course in the Theory of Groups, 2nd edn. (Springer, Berlin, 1995)
T.G. Room, P.B. Kirkpatrick, Miniquaternion Geometry (Cambridge University Press, Cambridge, 1971)
J.J. Rotman, An Introduction to the Theory of Groups, 4th edn. (Springer, Berlin, 1999)
W.R. Scott, Group Theory (Dover, New York, 1987)
M. Suzuki, Group Theory 1 & 2 (Springer, Berlin, 1982, 1986)
H. Wähling, Theorie der Fastkörper (Thales Verlag Essen, 1987)
H. Wielandt, Finite Permutation Groups (Academic Press, London, 1964)
J.A. Wolf, Spaces of Constant Curvature (AMS Chelsea, 2000)
H. Zassenhaus, The Theory of Groups (Chelsea, 1949)
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Lockhart, R. (2021). Near-Fields. In: The Theory of Near-Rings. Lecture Notes in Mathematics, vol 2295. Springer, Cham. https://doi.org/10.1007/978-3-030-81755-8_3
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