Cayley properties of merged Johnson graphs

Abstract

Extending earlier results of Godsil and of Dobson and Malnič on Johnson graphs, we characterise those merged Johnson graphs $J=J(n,k)_I$ which are Cayley graphs, that is, which are connected and have a group of automorphisms acting regularly on the vertices. We also characterise the merged Johnson graphs which are not Cayley graphs but which have a transitive group of automorphisms with vertex-stabilisers of order 2. Even though these merged Johnson graphs are all vertex-transitive, we show that only relatively few of them are Cayley graphs or have a transitive group of automorphisms with vertex-stabilisers of order 2.

Appendix: near-fields and sharply 2-transitive groups

The following summary of near-fields and their connection with sharply 2-transitive permutation groups is adapted to suit our purposes from those given by Cameron in [6, §1.12], by Dixon and Mortimer in [10, §7.6], and by Hall in [14, §20.7].

A near-field F satisfies all the usual field axioms, apart from possibly commutativity of multiplication $ab=ba$ and the left distributive axiom $a(b+c)=ab+ac$. (This concept should not be confused with that of a skew field, or division ring, in which only commutativity of multiplication is relaxed.) A near-field is a field if and only if multiplication is commutative.

In the finite case, near-fields are essentially equivalent to sharply 2-transitive permutation groups. Firstly, if F is any near-field then the affine group

$$\begin{aligned} AGL_1(F)=\{t\mapsto ta+b \mid a, b\in F, a\ne 0\} \end{aligned}$$

is a sharply 2-transitive group of transformations of F. The translations $t\mapsto t+b$ form a regular normal subgroup N, isomorphic to the additive group of F, while the transformations $t\mapsto ta\;(a\ne 0)$ form a complement, the stabiliser of 0, isomorphic to the multiplicative group $F^*=F\setminus \{0\}$. Conversely, any sharply 2-transitive permutation group G is a primitive Frobenius group, so if it is finite then it has an elementary abelian regular normal subgroup N (the Frobenius kernel), which can be identified with the set permuted. Then, the group structures of N and of the stabiliser $G_0$ of the zero element $0\in N$ provide the additive and multiplicative groups of a near-field F with underlying set N, such that G acts on F as $AGL_1(F)$. Thus, the classification of sharply 2-transitive finite groups is equivalent to that of finite near-fields; these classifications were achieved by Zassenhaus in [28, 29].

All but seven of the finite near-fields arise from a construction due to Dickson [9], in which the multiplicative group of a finite field is ‘twisted’ by field automorphisms to produce a non-commutative multiplication. Here, reversing the order used in [6, §1.12] and [10, §7.6], we will describe first the groups and then the near-fields.

If n is a prime power $p^e$ then for each d dividing e the field ${{\mathbb {F}}}_n$ has a (unique, cyclic) group of automorphisms $\varGamma$ of order d, generated by $\theta :t\mapsto t^q$ where $q=p^{e/d}$. The fixed field of $\varGamma$ is the subfield ${{\mathbb {F}}}_q$. Similarly, if

(a) d divides $n-1=q^d-1$,

then ${{\mathbb {F}}}_n^*$ has a (unique, cyclic) subgroup A of index d, consisting of its dth powers. We will use these two groups to form a group H of semilinear transformations of ${{\mathbb {F}}}_n$, fixing 0 and acting regularly on ${{\mathbb {F}}}_n^*$, so that we have a sharply 2-transitive group $G=N\rtimes G_0\le A\varGamma L_1({{\mathbb {F}}}_n)$ of degree n, where N is the translation group and $G_0=H$.

Suppose that

(b) $\{m(i)\mid i=0,\ldots , d-1\}$ is a complete set of residues mod (d) in $\mathbb Z$,

so that if $\omega$ is any generator for the cyclic group ${{\mathbb {F}}}_n^*$ then $\{\omega ^{m(i)}\mid i=0,\ldots , d-1\}$ is a set of coset representatives for A in ${{\mathbb {F}}}_n^*$. We now let each $g\in A\omega ^{m(i)}\subseteq {{\mathbb {F}}}_n^*$ induce the semilinear transformation

$$\begin{aligned} \tau _g:t\mapsto t^{\theta ^i}g=t^{q^i}g \end{aligned}$$

of ${{\mathbb {F}}}_n$, and we define $H:=\{\tau _g\mid g\in {{\mathbb {F}}}_n^*\}$. We need to ensure that H is a group under composition. If $h\in A\omega ^{m(j)}$ then (composing from left to right) we have

$$\begin{aligned} \tau _g \circ \tau _h : t \mapsto (t^{q^i}g)^{q^j}h=t^{q^{i+j}}g^{q^j}h, \end{aligned}$$

so for this to have the form $\tau _k$ for some $k\in {{\mathbb {F}}}_n^*$ we must ensure that $g^{q^j}h\in A\omega ^{m(i+j)}$. This will happen if our chosen residues m(i) satisfy

$$\begin{aligned} m(i+j)=q^jm(i)+m(j) \end{aligned}$$

for all i and j. An obvious solution for this identity is to take each

$$\begin{aligned} m(i)=1+q+q^2+\cdots +q^{i-1}=\frac{q^i-1}{q-1}. \end{aligned}$$

Then H is closed under composition, so it is a subgroup of $\varGamma L_1({{\mathbb {F}}}_n)$. It fixes 0, and is transitive on ${{\mathbb {F}}}_n^*$ since $\tau _g$ sends 1 to g for each $g\in {{\mathbb {F}}}_n^*$. Since $|H|=|{{\mathbb {F}}}_n^*|$ we deduce that H acts regularly on ${{\mathbb {F}}}_n^*$, so that

$$\begin{aligned} G:=\{t\mapsto t\tau _g+b\mid g\in {{\mathbb {F}}}_n^*, b\in {{\mathbb {F}}}_n\} \end{aligned}$$

acts on ${{\mathbb {F}}}_n$ as a sharply 2-transitive subgroup of $A\varGamma L_1({{\mathbb {F}}}_n)$.

Note there is an epimorphism $H\rightarrow \varGamma , g\mapsto \theta ^i$ where $g\in A\omega ^{m(i)}$. The kernel is A, so (like ${{\mathbb {F}}}_n^*$) H is an extension of a normal subgroup A by $\varGamma$. Indeed, H and ${{\mathbb {F}}}_n^*$ induce the same group structure on their subgroup A, and also on its quotient group, even though H, being nonabelian, is not isomorphic to ${{\mathbb {F}}}_n^*$ if $d>1$.

Having motivated this definition of m(i), we need to choose q and d carefully so that conditions (a) and (b) are satisfied. Elementary number theory (see [23, Theorem 6.4] or [29] for details) shows that the following is a sufficient condition for this:

(c) if r divides d, where r is prime or $r=4$, then r divides $q-1$.

The near-field F corresponding to G has the same underlying set and additive structure as ${{\mathbb {F}}}_n$, but multiplication (of nonzero elements g and h) reflects composition of the corresponding elements of $\tau _g, \tau _h\in H$, so that the product of g and h in F is given by $g\circ h=k=g^{q^j}h$ where $h\in A\omega ^{m(j)}$ (and hence the centre of F is ${{\mathbb {F}}}_q$). The group G defined above can now be identified with the group $AGL_1(F)$ of affine transformations of this near-field F.

Dickson near-fields F of order $n\equiv 3$ mod (4) appear in Theorem 1(1). In such cases, e and hence d are odd, so H has a unique subgroup $H^2$ of index 2, consisting of those $\tau _g$ such that g is a square in ${{\mathbb {F}}}_n^*$. Extending $H^2$ by the translation group gives the unique subgroup $AHL_1(F)$ of index 2 in $AGL_1(F)$. The involution $-1\in {{\mathbb {F}}}_n^*$ is in A, so $\tau _{-1}$ acts on F as $t\mapsto -t$; the involution $f:t\mapsto t\tau _{-1}+1$ in $AGL_1(F)$ thus preserves the 2-element subset $\{0, 1\}$ of F, and hence generates its stabiliser. Since $n\equiv 3$ mod (4), $-1$ is a non-square in ${{\mathbb {F}}}_n^*$, so $AHL_1(F)$ does not contain f and hence acts regularly on 2-element subsets of F.

In addition to the Dickson near-fields described above, there are seven exceptional finite near-fields F: together with the Dickson near-fields, they appear in Theorem 2(1). They have order $n=p^2$ and thus correspond to sharply 2-transitive groups $G=AGL_1(F)$ of degree n, for the primes $p=5$, 7, 11 (twice), 23, 29 and 59. In each case, G is a subgroup of $AGL_2(p)$ containing the translation group, and $G_0$ ($\cong F^*$) is a subgroup of $GL_2(p)$ acting regularly on nonzero vectors. When $p=5$, 7 or 11 one can take $G_0$ to be the binary tetrahedral, binary octahedral or binary icosahedral group $2T\cong SL_2(3)$, $2O\cong 2.S_4^-$ or $2I\cong SL_2(5)$; for $p=11$ (again), 23, 29 or 59 one can take $G_0=2T\times C_5$, $2O\times C_{11}$, $2I\times C_7$ or $2I\times C_{29}$, with the cyclic direct factor consisting of scalar matrices.