Namespaces
Variants
Actions

Difference between revisions of "Unital"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX done)
Line 1: Line 1:
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u110/u110030/u1100301.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u110/u110030/u1100302.png" />-design (cf. also [[Block design|Block design]]). It arose originally as the set of self-conjugate points and non-self-conjugate lines in a unitary [[Polarity|polarity]] in a Desarguesian projective plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u110/u110030/u1100303.png" /> (cf. [[Desargues geometry|Desargues geometry]]), in which case it has an automorphism group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u110/u110030/u1100304.png" /> with associated simple group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u110/u110030/u1100305.png" /> (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u110/u110030/u1100306.png" />); see [[#References|[a3]]]. This type is known as a classical or Hermitian unital. The points can be considered as points of the curve with equation
+
A $2$-$(q^3+1,q+1,1)$-design (cf. also [[Block design]]). It arose originally as the set of self-conjugate points and non-self-conjugate lines in a unitary [[polarity]] in a Desarguesian projective plane $\mathrm{PG}(2,q^2)$ (cf. [[Desargues geometry]]), in which case it has an automorphism group $\mathrm{P}\Gamma\mathrm{U}(3,q^2)$ with associated simple group $\mathrm{PSU}(3,q^2)$ (when $q>2$); see [[#References|[a3]]]. This type is known as a classical or Hermitian unital. The points can be considered as points of the curve with equation
 +
$$
 +
X^{q+1} + Y^{q+1} + Z^{q+1} = 0
 +
$$
 +
whose coordinates lie in the field $\mathrm{GF}(q^2)$. The design was first explicitly constructed by R.C. Bose [[#References|[a2]]]. In fact, such a polarity and hence a unital exists in a non-Desarguesian plane constructed from a finite commutative semi-field with an involution [[#References|[a6]]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u110/u110030/u1100307.png" /></td> </tr></table>
+
A unital with $q=6$ has been constructed by R. Mathon [[#References|[a5]]] and by S. Bagchi and B. Bagchi [[#References|[a1]]]. This shows that a unital of order $q$ cannot necessarily be embedded in a plane of order $Q$. It had in fact been shown by H. Lüneberg [[#References|[a4]]] that another class of unitals, the Ree unitals, having an associated simple automorphism group $G$, cannot be embedded in a projective plane in such a way that $G$ is induced by a collineation group of the plane.
  
whose coordinates lie in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u110/u110030/u1100308.png" />. The design was first explicitly constructed by R.C. Bose [[#References|[a2]]]. In fact, such a polarity and hence a unital exists in a non-Desarguesian plane constructed from a finite commutative semi-field with an involution [[#References|[a6]]].
+
A class of unitals other than the Hermitian ones and those embeddable in $\mathrm{PG}(2,q^2)$ are the [[Buekenhout–Metz unital]]s.
  
A unital with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u110/u110030/u1100309.png" /> has been constructed by R. Mathon [[#References|[a5]]] and by S. Bagchi and B. Bagchi [[#References|[a1]]]. This shows that a unital of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u110/u110030/u11003010.png" /> cannot necessarily be embedded in a plane of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u110/u110030/u11003011.png" />. It had in fact been shown by H. Lüneberg [[#References|[a4]]] that another class of unitals, the Ree unitals, which have an associated simple automorphism group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u110/u110030/u11003012.png" />, cannot be embedded in a projective plane in such a way that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u110/u110030/u11003013.png" /> is induced by a collineation group of the plane.
+
====References====
 +
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> S. BagchiB. Bagchi,  "Designs from pairs of finite fields I. A cyclic unital $U(6)$ and other regular Steiner $2$-designs"  ''J. Combin. Th. A'' , '''52'''  (1989)  pp. 51–61</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  R.C. Bose,  "On the application of finite projective geometry for deriving a certain series of balanced Kirkman arrangements" , ''Golden Jubilee Commemoration Volume, 1958-1959'' , Calcutta Math. Soc. (1959)  pp. 341–354</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  J.W.P. Hirschfeld,  "Projective geometries over finite fields" , Oxford Univ. Press  (1979)</TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top"> H. Lüneburg,   "Some remarks concerning Ree groups of type $(G_2)$"  ''J. Algebra'' , '''3'''  (1966)  pp. 256–259</TD></TR>
 +
<TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Mathon,  "Constructions of cyclic 2-designs" ''Ann. Discrete Math.'' , '''34'''  (1987)  pp. 353–362</TD></TR>
 +
<TR><TD valign="top">[a6]</TD> <TD valign="top">  F. Piper,  "Unitary block designs" R.J. Wilson (ed.) , ''Graph Theory and Combinatorics'' , ''Research Notes in Mathematics'' , '''34''' , Pitman  (1979)  pp. 98–105</TD></TR>
 +
</table>
  
A class of unitals other than the Hermitian ones and those embeddable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u110/u110030/u11003014.png" /> are the Buekenhout–Metz unitals (cf. [[Buekenhout–Metz unital|Buekenhout–Metz unital]]).
+
{{TEX|done}}
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Bagchi,  B. Bagchi,  "Designs from pairs of finite fields I. A cyclic unital <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u110/u110030/u11003015.png" /> and other regular Steiner <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u110/u110030/u11003016.png" />-designs"  ''J. Combin. Th. A'' , '''52'''  (1989)  pp. 51–61</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.C. Bose,  "On the application of finite projective geometry for deriving a certain series of balanced Kirkman arrangements" , ''Golden Jubilee Commemoration Volume, 1958-1959'' , Calcutta Math. Soc.  (1959)  pp. 341–354</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.W.P. Hirschfeld,  "Projective geometries over finite fields" , Oxford Univ. Press  (1979)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Lüneburg,  "Some remarks concerning Ree groups of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u110/u110030/u11003017.png" />"  ''J. Algebra'' , '''3'''  (1966)  pp. 256–259</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Mathon,  "Constructions of cyclic 2-designs"  ''Ann. Discrete Math.'' , '''34'''  (1987)  pp. 353–362</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  F. Piper,  "Unitary block designs"  R.J. Wilson (ed.) , ''Graph Theory and Combinatorics'' , ''Research Notes in Mathematics'' , '''34''' , Pitman  (1979)  pp. 98–105</TD></TR></table>
 

Revision as of 19:50, 7 August 2016

A $2$-$(q^3+1,q+1,1)$-design (cf. also Block design). It arose originally as the set of self-conjugate points and non-self-conjugate lines in a unitary polarity in a Desarguesian projective plane $\mathrm{PG}(2,q^2)$ (cf. Desargues geometry), in which case it has an automorphism group $\mathrm{P}\Gamma\mathrm{U}(3,q^2)$ with associated simple group $\mathrm{PSU}(3,q^2)$ (when $q>2$); see [a3]. This type is known as a classical or Hermitian unital. The points can be considered as points of the curve with equation $$ X^{q+1} + Y^{q+1} + Z^{q+1} = 0 $$ whose coordinates lie in the field $\mathrm{GF}(q^2)$. The design was first explicitly constructed by R.C. Bose [a2]. In fact, such a polarity and hence a unital exists in a non-Desarguesian plane constructed from a finite commutative semi-field with an involution [a6].

A unital with $q=6$ has been constructed by R. Mathon [a5] and by S. Bagchi and B. Bagchi [a1]. This shows that a unital of order $q$ cannot necessarily be embedded in a plane of order $Q$. It had in fact been shown by H. Lüneberg [a4] that another class of unitals, the Ree unitals, having an associated simple automorphism group $G$, cannot be embedded in a projective plane in such a way that $G$ is induced by a collineation group of the plane.

A class of unitals other than the Hermitian ones and those embeddable in $\mathrm{PG}(2,q^2)$ are the Buekenhout–Metz unitals.

References

[a1] S. Bagchi, B. Bagchi, "Designs from pairs of finite fields I. A cyclic unital $U(6)$ and other regular Steiner $2$-designs" J. Combin. Th. A , 52 (1989) pp. 51–61
[a2] R.C. Bose, "On the application of finite projective geometry for deriving a certain series of balanced Kirkman arrangements" , Golden Jubilee Commemoration Volume, 1958-1959 , Calcutta Math. Soc. (1959) pp. 341–354
[a3] J.W.P. Hirschfeld, "Projective geometries over finite fields" , Oxford Univ. Press (1979)
[a4] H. Lüneburg, "Some remarks concerning Ree groups of type $(G_2)$" J. Algebra , 3 (1966) pp. 256–259
[a5] R. Mathon, "Constructions of cyclic 2-designs" Ann. Discrete Math. , 34 (1987) pp. 353–362
[a6] F. Piper, "Unitary block designs" R.J. Wilson (ed.) , Graph Theory and Combinatorics , Research Notes in Mathematics , 34 , Pitman (1979) pp. 98–105
How to Cite This Entry:
Unital. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unital&oldid=15809
This article was adapted from an original article by J.W.P. Hirschfeld (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article