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  "body": "\n## Definition\n\nLet $n \\ge 2$ be an integer.\nA *unital of order $n$* is a [[block design|4JCSS2Z1BA]] $(X, \\mathcal{B})$ such that:\n\n1. the point set $X$ has $n^3 + 1$ elements;\n2. every block $B \\in \\mathcal{B}$ has $n + 1$ elements;\n3. any two distinct points of $X$ lie in exactly one common block.\n\n## Remarks\n\nEquivalently, a unital of order $n$ is a [[Steiner system|7BPSC4T4M3]] $S(2, n + 1, n^3 + 1)$.\n\nLet $\\Pi$ be a [[projective plane|DANFNN8C4Y]] of order $n^2$.\nA unital is *embedded* in $\\Pi$ if $X$ is a set of points of $\\Pi$ and the blocks are the intersections of $X$ with lines that have size $n + 1$.\nIn that case every line of $\\Pi$ meets $X$ in $1$ or $n + 1$ points.\n\nExample: let $q$ be a prime power and let $\\F_{q^2}$ be the [[finite field|0ZXZ82FTX2]] with $q^2$ elements.\nIn $\\mathrm{PG}(2, q^2)$, the Hermitian curve $x^{q+1} + y^{q+1} + z^{q+1} = 0$, the set of isotropic points of a non-degenerate [[Hermitian form|48P4A1CX85]], has $q^3 + 1$ points.\nEach line meets it in $1$ or $q + 1$ points, so with the secant lines as blocks it is a unital of order $q$.\n\nNon-example: in the Fano plane, blocks have size $3$ and any two points lie in exactly one block.\nBut it has $7$ points rather than $2^3 + 1 = 9$, so clause 1 fails.\n",
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