View source for Unital

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]]].

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.

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

====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 $U(6)$ 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>

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