Correction to: Flag-transitive block designs and unitary groups

1 Correction to: Monatsh Math https://doi.org/10.1007/s00605-020-01421-8

Abstract

In this note, we cover a gap in the proof of [2, Proposition 4.3]. In conclusion, Theorem 1.1 in [2] is revisited: if $\mathcal {D}$ is a 2-design with $\gcd (r, \lambda )=1$ and G is a flag-transitive almost simple automorphism group of $\mathcal {D}$ whose socle is PSU (n, q) with $(n, q)\ne (3, 2)$, then $\mathcal {D}$ belongs to one of the three infinite families of Hermitian unitals, Witt–Bose–Shrikhande spaces and 2-designs with parameters (q$^{3}+1$, q, q$-1$), or it is isomorphic to a design with parameters (6, 3, 2), (7, 3, 1), (8, 4, 3), (10, 6, 5), (11, 5, 2) or (28, 7, 2).

Keywords 2-Design, Flag-transitive, Automorphism group, Almost simple group, Unitary group

Mathematics Subject Classification 05B05 $\cdot$ 05E18$\cdot$ 20D05

2 Introduction

A 2-design $\mathcal {D}$ with parameters $(v, k, \lambda )$ is a pair $(\mathcal {P}, \mathcal {B})$ with a set $\mathcal {P}$ of v points and a set $\mathcal {B}$ of b blocks such that each block is a k-subset of $\mathcal {P}$ and each two distinct points are contained in $\lambda$ blocks. We say $\mathcal {D}$ is nontrivial if $2< k < v-1$, and symmetric if $v = b$. Each point of $\mathcal {D}$ is contained in exactly r blocks which is called the replication number of $\mathcal {D}$. A flag of $\mathcal {D}$ is a point-block pair $(\alpha , B)$ such that $\alpha \in B$. An automorphism of a 2-design $\mathcal {D}$ is a permutation of the points permuting the blocks and preserving the incidence relation. The full automorphism group ${\mathrm {Aut}}(\mathcal {D})$ of $\mathcal {D}$ is the group consisting of all automorphisms of $\mathcal {D}$. For $G\leqslant {\mathrm {Aut}}(\mathcal {D})$, G is called flag-transitive if G acts transitively on the set of flags and G is said to be point-primitive if it is primitive on $\mathcal {P}$. In this note, we cover a gap in the proof of [2, Proposition 4.3]. Therefore, we correct Theorem 1.1 in [2] as below:

Theorem 1.1

Let $\mathcal {D}$ be a nontrivial 2-design with $\gcd (r, \lambda )=1$, and let $\alpha$ be a point of $\mathcal {D}$. Suppose that G is an automorphism group of $\mathcal {D}$ whose socle is $X={\mathrm {PSU}}(n, q)$ with $(n, q)\ne (3, 2)$. If G is flag-transitive, then $\lambda \in$ $\{1, 2, 3, 5\}$ and v, k, $\lambda$, $X_{\alpha }$ and X are as in one of the lines in Table 1 or one of the following holds:

• (a) $\mathcal {D}$ is a Witt–Bose–Shrikhande space with parameters $(2^{n-1}(2^{n}-1), 2^{n-1},1)$ and X is ${\mathrm {PSU}}(2, 2^n)$ with $n\geqslant 3$;

• (b) $\mathcal {D}$ is a Hermitian unital $\mathcal {U}_{H}(q)$ with parameters $(q^{3}+1,q+1,1)$ and X is ${\mathrm {PSU}}(3, q)$;

• (c) $\mathcal {D}$ is a 2-design with parameters $(q^{3}+1,q,q-1)$ and X is ${\mathrm {PSU}}(3, q)$, and the point set of $\mathcal {D}$ is the point set of a Hermitian unital $\mathcal {U}_{H}(q)$ and the block set is $(\ell {\setminus }\{\gamma \})^{G}$ where $\ell$ is a line of $\mathcal {U}_{H}(q)$ and $\gamma \in \ell$.

Remark 1.1

We remark here that the class $\mathcal {C}_{5}$ should be excluded from [2, Lemma 3.11] when H is of type ${\mathrm {GU}}_{n}(q_{0})$ with $q=q_{0}^{t}$ and t odd prime. However, this change does not affect the proof of [2, Proposition 4.3] as the large subgroup condition in [2, Lemma 3.6] implies in this case that $t=3$ which was handled in [2, Propositions 4.1 and 4.3].

It is worth noting by [6] that there is a general construction method for 2-designs from linear space: For a 2-(v, k, 1) design $\mathcal {S}= (\mathcal {P}, \mathcal {L})$ with $k \geqslant 3$, let $\mathcal {B}=\{\ell {\setminus }\{\alpha \}\mid \ell \in \mathcal {L}, \alpha \in \ell \}$ and $\mathcal {D}(\mathcal {S}) = (\mathcal {P},\mathcal {B})$. Then [6, Proposition 4.1] implies that $\mathcal {D}(\mathcal {S})$ is a 2-$(v,k-1,k-2)$ design, and moreover, that G is flag-transitive on $\mathcal {D}(\mathcal {S})$ whenever $G \leqslant {\mathrm {Aut}}(\mathcal {S})$ is flag-transitive on $\mathcal {S}$ and induces a 2-transitive action on each line of $\mathcal {S}$. Therefore, the design in Theorem 1.1 can be obtained in this way by taking $\mathcal {S}$ as the Hermitian unital $\mathcal {U}_{H}(q)$.

Table 1 Some nontrivial 2-design with $\gcd (r,\lambda )=1$

3 Proof of Theorem 1.1

In this section, we prove Proposition 2.1 below, and this together with [2, Proposition 4.2] will prove Theorem 1.1. In order to prove Proposition 2.1, we first need to introduce the Hermitian unitals. Here, we follow the same terminology as in [8] with a few exceptions in our notation.

Let $q =p^a >2$ with p a prime. The mapping $x\mapsto x^{q}$ is an automorphism of the Galois field $\mathbb {F}_{q^{2}}$, which we will write as $x^{q}=\bar{x}$ occasionally. The Galois field $\mathbb {F}_{q}$ is then the fixed field of this automorphism. Let V be a three-dimensional vector space over $\mathbb {F}_{q^{2}}$ and $\varphi$ a nondegenerate $\sigma$-Hermitian form on V. The full unitary group $\Gamma {\mathrm {U}}(3,q)$ consists of those semilinear transformations of V that induce a collineation of ${\mathrm {PG}}(2,q^{2})$ which commutes with $\varphi$. The general unitary group ${\mathrm {GU}}(3, q ) = \Gamma {\mathrm {U}}(3,q)\cap {\mathrm {GL}}(3,q^{2})$ is the group of nonsingular linear transformations of V leaving $\varphi$ invariant. The projective unitary group ${\mathrm {PGU}}(3,q)$ is the quotient group ${\mathrm {GU}}(3, q)/Z$, where $Z = \{aI \mid a \in \mathbb {F}_{q^{2}},\ a^{q+1} = 1\}$ is the center of ${\mathrm {GU}}(3,q)$ and I the identity transformation. The special projective unitary group ${\mathrm {PSU}}(3, q )$ is the quotient group ${\mathrm {SU}}(3, q )/(Z \cap {\mathrm {SU}}(3, q ))$, where ${\mathrm {SU}}(3, q )$ is the subgroup of ${\mathrm {GU}}(3,q)$ consisting of linear transformations of unit determinant. The group ${\mathrm {PSU}}(3,q)$ is equal to ${\mathrm {PGU}}(3,q)$ if 3 is not a divisor of $q+1$, and is a subgroup of ${\mathrm {PGU}}(3,q)$ of index 3 otherwise. It is well-known that the automorphism group of ${\mathrm {PSU}}(3, q)$ is equal to ${\mathrm {P}}\Gamma {\mathrm {U}}(3, q):= {\mathrm {PGU}}(3,q) \langle \sigma _{p} \rangle$, where $\sigma _{p}:x\mapsto x^{p}$ is the Frobenius map. By [8, Lemma 4.1], we choose an appropriate basis $\{e_1, e_{2}, e_{3}\}$ for V with corresponding Hermitian matrix of $\varphi$ by

$$\begin{aligned} \left( \begin{array}{ccc} 0 &{}\quad 0 &{}\quad 1 \\ 0 &{}\quad 1 &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 0 \\ \end{array} \right) . \end{aligned}$$

If $u =(x_{1},x_{2},x_{3})$ and $v=(y_{1},y_{2},y_{3})$ are vectors in V, then $\varphi (u,v)=x_{1}y_{3}^{q}+x_{2}y_{2}^{q}+x_{3}y_{1}^{q}$. A vector $u\in V$ is called isotropic if $\varphi (u,u)=0$ and nonisotropic otherwise. Let

$$\begin{aligned} \mathcal {P}=\{\langle 0,0,1 \rangle , \langle 1,a,b \rangle \mid a,b\in \mathbb {F}_{q^{2}}\text { and } a^{q+1}+b+b^{q}=0\}, \end{aligned}$$

(1)

where $\langle a,b,c \rangle$ denotes the 1-dimensional subspace of V spanned by $(a,b,c)\in V$. The elements of $\mathcal {P}$ are called the absolute points. It is well-known that $|\mathcal {P}| = q^{3} + 1$, ${\mathrm {PSU}}(3, q)$ is 2-transitive on $\mathcal {P}$, and ${\mathrm {P}}\Gamma {\mathrm {U}}(3,q)$ leaves $\mathcal {P}$ invariant. Denote

$$\begin{aligned} \infty :=\langle 0,0,1\rangle \text { and } 0:=\langle 1,0,0 \rangle , \end{aligned}$$

and set

$$\begin{aligned} \varDelta :=\{\langle 1,0,b \rangle \mid b\in \mathbb {F}_{q^{2}}\text { and } b+b^{q}=0\}, \end{aligned}$$

(2)

and let H be the point-stabiliser of $\infty$ in $X={\mathrm {PSU}}(3,q)$, that is to say, $H=X_{\infty }$. By [8], we have the following information about these groups and their actions on $\mathcal {P}$:

• (a) $H=QL$, where Q is a normal subgroup of H of order $q^{3}$ which acts regularly on $\mathcal {P}{\setminus }\{\infty \}$ and $L=X_{\infty ,0}$ which is a cyclic subgroup of H of order $(q^{2}-1)/\gcd (3,q+1)$;

• (b) $L=X_{\infty ,0}$ has two trivial orbits $\{0\}$, $\{\infty \}$, one nontrivial orbit $\varDelta {\setminus }\{0\}=\{\langle 1,0,b \rangle \mid 0\ne b\in \mathbb {F}_{q^{2}}\text { and } b+b^{q}=0\}$ of length $q-1$, and its remaining nontrivial orbits are of length $(q^{2}-1)/\gcd (3,q+1)$;

• (c) $P=Z(Q)=[Q,Q]$ is a subgroup of Q of order q fixing $\infty$ and acting transitively on $\varDelta$ defined in (2);

• (d) $PL=X_{\infty ,\ell (\infty )}$ is transitive on $\varDelta$ and it is of order $q(q^{2}-1)/\gcd (3,q+1)$, where $\ell (\infty )=\{\infty \}\cup \varDelta$, that is to say,

  $$\begin{aligned} \ell (\infty )=\{\infty \}\cup \{\langle 1,0,b \rangle \mid b\in \mathbb {F}_{q^{2}}, \ b+b^{q}=0\}; \end{aligned}$$

  (3)

The Hermitian unital $\mathcal {U}_{H}(q)$ is defined to be the block design with the point set $\mathcal {P}$ in which a subset of $\mathcal {P}$ is a block (called a line) precisely when it is the set of absolute points contained in some $\langle u,v \rangle$. We know by [7, 8, 11] that $\mathcal {U}_{H}(q)$ is a linear space with $q^{3}+1$ points, $q^{2}(q^{2}- q+1)$ lines, $q+1$ points in each line, and $q^{2}$ lines on each point. It was proved in [8, 11] that ${\mathrm {Aut}}(\mathcal {U}_{H}(q)) = {\mathrm {P}}\Gamma {\mathrm {U}}(3, q)$. Thus, every G with $X={\mathrm {PSU}}(3,q)\leqslant G \leqslant {\mathrm {P}}\Gamma {\mathrm {U}}(3,q)$ acts 2-transitively on the point set of $\mathcal {U}_{H}(q)$. This implies that G is also block-transitive and flag-transitive on $\mathcal {U}_{H}(q)$. A line of ${\mathrm {PG}}(2,q^{2})$ contains either one absolute point or $q+1$ absolute points. In the latter case, the set of such $q+1$ absolute points is a line of $\mathcal {U}_{H}(q)$, and all lines of $\mathcal {U}_{H}(q)$ are of this form. In particular, $\ell (\infty )$ defined in (3) is a line of $\mathcal {U}_{H}(q)$ containing $\infty$ (see [8, Lemma 2.5]). Moreover, the line stabiliser $X_{\ell (\infty )}$ is transitive on $\ell (\infty )$ and $P\leqslant X_{\infty ,\ell (\infty )}$ is transitive on $\ell (\infty ){\setminus }\{\infty \}$, and hence $X_{\ell (\infty )}$ is 2-transitive on $\ell (\infty )$. Since X is flag-transitive, for each line $\ell$ of $\mathcal {U}_{H}(q)$, we conclude that $X_{\ell }$ is 2-transitive on $\ell$.

Suppose now that $B=\ell (\infty ){\setminus }\{0\}$. The information given above are useful to observe that $X_{\infty ,B}=X_{\infty ,0}$ and $X_{B}\leqslant X_{\ell (\infty )}$, and so $X_{B}=X_{0,B}$ is a subgroup of index $q+1$ in $X_{\ell (\infty )}$ and $|X_{B}:X_{\infty ,B}|=q$. Note that X is 2-transitive on $\mathcal {P}$. If $\mathcal {B}=B^{X}$, then $(\mathcal {P},\mathcal {B})$ is a 2-design with parameters $(q^3+1,q,q-1)$, and hence this gives an explicit construction for the design that appears in Theorem 1.1(c).

We are now ready to revisit Proposition 4.3 in [2], and prove Proposition 2.1 below. In what follows, we frequently use the results mentioned above about the Hermitian unitals and their automorphism groups.

Proposition 2.1

Let $\mathcal {D}$ be a nontrivial 2-design with $\gcd (r, \lambda )=1$. Suppose that G is an automorphism group of $\mathcal {D}$ whose socle is $X={\mathrm {PSU}}(n, q)$ with $n\geqslant 3$ and $(n, q)\ne (3, 2)$. If G is flag-transitive, then X is ${\mathrm {PSU}}(3, q)$, and one of the following holds:

• (a) $\mathcal {D}$ is a Hermitian unital with parameters $(q^{3}+1,q+1,1)$;

• (b) $\mathcal {D}$ is a 2-design with parameters $(q^{3}+1,q,q-1)$, and the point set of $\mathcal {D}$ is the point set of a Hermitian unital $\mathcal {U}_{H}(q)$ and the block set is $(\ell {\setminus }\{\gamma \})^{G}$ where $\ell$ is a line of $\mathcal {U}_{H}(q)$ and $\gamma \in \ell$.

Proof

Suppose that $H=G_{\alpha }$ with $\alpha$ a point of $\mathcal {D}$. If H is not a parabolic subgroup $P_{m}$, then we follow the same argument as in [2, Proposition 4.3] which leads to no possible parameters. Therefore, considering Remark 1.1, we only need to deal with the case where H is isomorphic to $P_{m}$, for some $2m\leqslant n$. In this case, by the same argument as in [2, Proposition 4.3], the inequality $v<r^{2}$ restricts to the case where $n=3$, that is to say, $X={\mathrm {PSU}}(3,q)$ and $H\cap X\cong \, ^{\hat{}}q^{3}(q^{2}-1)$ in which case $v=q^{3}+1$. If $\lambda =1$, then by [10], $\mathcal {D}$ is a Hermitian unital as in part (a). Suppose now that $\lambda >1$. Here, X acts 2-transitively on the point set of $\mathcal {D}$, and this action is permutationally isomorphic to the action of X on the set $\mathcal {P}$ as in (1). Therefore, without loss of generality, we can identify the point set of $\mathcal {D}$ with $\mathcal {P}$, and take $\alpha :=\infty$. Since $\gcd (r,\lambda )=1$, [5, 1.2.8] implies that X is flag-transitive, and hence we can also assume that $G=X$, and so $H=X_{\infty }\cong \, ^{\hat{}}q^{3}(q^{2}-1)$. Let B be a block containing $\infty$, and let $\ell :=\ell (\infty )$ be a line in $\mathcal {U}_{H}(q)$ passing through $\infty$. Since r divides $v-1=q^{3}$ where $q=p^{3a}$, it follows that $r=p^{t}$, for some $t\leqslant 3a$. Since also $b=rv/k$, we have that $|X_{B}|=|X|/b=kp^{3a-t}(q^{2}-1)$. By inspecting the maximal subgroups of X from [3, Table 8.5], we then conclude that $X_{B}$ is contained in $X_{\ell }$ which is isomorphic to $^{\hat{}}{\mathrm {GU}}_{2}(q)$. Since $X_{B}$ is contained in a maximal subgroup M of $X_{\ell }$ and $X_{\ell }$ is 2-transitive on $\ell$, M is a point-stabiliser of $X_{\ell }$. By possibly replacing B with its conjugate, we can assume that $X_{B}\leqslant X_{0,\ell }$. Thus $X_{\infty ,B}$ is contained in $X_{\infty ,0,\ell }=X_{\infty ,0}$. Since $bk=vr=p^t(q^{3}+1)$ and $bk=|X:X_{B}|\cdot |X_{B}:X_{\infty ,B}|=|X:X_{\infty ,B}|$, we conclude that $|X_{\infty ,B}|=p^{3a-t}(q^{2}-1)/d$. Recall that $X_{\infty ,B}\leqslant X_{\infty ,0}$ and $X_{\infty ,0}$ is a cyclic group of order $(q^{2}-1)/d$. Therefore, $X_{\infty ,B}=X_{\infty ,0}$. We know that $|X_{0,\ell }:X_{\infty ,0}|=q$. Since $X_{0,B}$ is contained in $X_{0,\ell }$, it follows that $k=|X_{B}:X_{\infty ,B}|=|X_{B}:X_{\infty ,0}|\leqslant |X_{0,\ell }:X_{\infty ,0}|=q$, that is to say, $k\leqslant q$. Recall that $X_{\infty ,B}=X_{\infty ,0}$. Then $X_{\infty ,0}$ fixes B, and so $B{\setminus }\{\infty \}$ is a union of nontrivial $X_{\infty ,0}$-orbits. We know that $X_{\infty ,0}$ fixes $\infty$ and 0, and it has one nontrivial orbit of length $q-1$ and its remaining nontrivial orbits are of length $(q^{2}-1)/d$. Since $k\leqslant q$, we conclude that $B{\setminus }\{\infty \}$ is the nontrivial $X_{\infty ,0}$-orbit $\ell {\setminus }\{\infty ,0\}$ of length $q-1$. Therefore, $B=\ell {\setminus }\{0\}$. Indeed, $B=\{\infty \}\cup (\varDelta {\setminus }\{0\})$, where $\varDelta$ is as in (2). This implies that $k=q$, $b=q^{2}(q^{3}+1)$ and $\lambda =q-1$. In conclusion, $\mathcal {D}$ is a 2-design with parameters $(q^{3}+1,q,q-1)$. If X fixes 0 and $\ell$, then it fixes $\ell {\setminus }\{0\}$. Thus $X_{0,\ell }\leqslant X_{B}$, and since $X_{0,\ell }$ is transitive on $B=\ell {\setminus }\{0\}$, it follows that $X_{B}$ is transitive on $B=\ell {\setminus }\{0\}$, and hence X is flag-transitive. Therefore, $\mathcal {D}$ is a 2-design with parameters $(q^{3}+1,q,q-1)$ whose points are the points of $\mathcal {U}_{H}(q)$ and $\mathcal {B}=B^X$, where $B=\ell {\setminus }\{0\}$ with $\ell$ a line of $\mathcal {U}_{H}(q)$. $\square$