The minimal degree of a permutation group G is defined as the minimal number of non-fixed points of a non-trivial element of G. In this paper, we show that if G is a transitive permutation group of degree n having no non-trivial normal 2-subgroups such that the stabilizer of a point is a 2-group, then the minimal degree of G is at least . The proof depends on the classification of finite simple groups.
Given a finite group acting on a finite set Ω and , we let
denote the set of fixed points of g, and we let . Then the parameter
is called the minimal degree of the permutation group G. Minimal degree (and a related parameter , sometimes called the fixity of G) has been the focus of intense study by several authors with most work concentrating on proving upper and lower bounds on the minimal degree of primitive permutation groups and permutation actions of classical groups (see [2, 5, 13, 20, 15], to name a few). Far fewer papers deal with other permutation groups (see [18, 23]).
The aim of this paper is to consider a lower bound on the minimal degree of a transitive permutation group G whose point stabilizer is a 2-group and such that (where by we denote the largest normal 2-subgroup of G). In particular, we prove the following.
Let G be a transitive permutation group on a set Ω with such that the point stabilizer is a 2-group. Then .
Our motivation to prove this result stems from the investigation of the minimal degree of permutation groups appearing as arc-transitive automorphism groups of finite connected (possibly directed) graphs or, equivalently, permutation groups admitting a connected suborbit, where by a connected suborbit of a transitive permutation group we mean an orbit Σ of the point stabilizer acting on , such that the directed graph with vertex-set Ω and edge-set is connected. Note that a transitive permutation group is primitive if and only if all of its non-trivial suborbits are connected. In view of the numerous results regarding the minimal degree of permutation groups, it is thus natural to relax the condition of primitivity to that of an existence of a single connected suborbit. In fact, based on the available computational data [10, 11, 25, 26], one can see that, under some additional assumptions on the length and the self-pairedness of the connected suborbit (and modulo some well-defined exceptions), lower bounds on the minimal degree holding for primitive permutation groups seem to extend to this more general situation. Theorem 1.1 represents a crucial step in a recent result  stating that, apart from a well-understood infinite family and a finite set of examples, every transitive permutation group admitting a connected suborbit of length at most 2 or a self-paired connected suborbit of length at most 4 satisfies (a suborbit Σ is self-paired if, for every edge of the corresponding directed graph, also the pair is an edge of that directed graph). Arc-transitive automorphism groups of graphs having point stabilizer a 2-group have already cropped up a few times in the literature; see for instance [31, 32].
We would also like to point out that Guralnick and Magaard  (building on earlier work of Lawther, Liebeck and Saxl [20, 21]) have proved that, except for an explicit list of exceptions, the minimal degree of every primitive permutation group of degree n is at least . Our proof of Theorem 1.1 quickly reduces to the case that the group G under consideration is quasiprimitive. In this sense, our result can be thought of as an attempt to extend the result of Guralnick and Magaard to the case of quasiprimitive groups.
We use fairly standard notation. Given a set Ω, we denote by and the symmetric and the alternating group on Ω. When the domain Ω is irrelevant or clear from the context, we write and for the symmetric and alternating group of degree n. Given a permutation , we write for the fixed-point ratio of g, that is,
Using this notation, Theorem 1.1 is equivalent to the following.
Let G be a transitive permutation group on a set Ω with such that the point stabilizer is a 2-group. Then for every non-identity element g of G.
Sometimes, for instance in Lemma 2.1, we need slightly more general notation: given a subset Δ of Ω, we let .
A subgroup G of is said to be semiregular if the identity is the only element of G fixing some point of Ω. Let G be a group, and let H be a subgroup of G; we denote by the set of right cosets of H in G. Recall that G acts transitively on by right multiplication. The normalizer of H in G is denoted by .
2 Proof of Theorem 1.1
We begin with two useful lemmas and then proceed to the proof of Theorem 1.1.
Let X be a group acting transitively on Ω, let Σ be a system of imprimitivity for X in its action on Ω, and let . Then .
This is clear because, if x fixes a block , then and hence . ∎
Let X be a group acting on a set Ω, let Y be a normal subgroup of X, let , and let . Then
where is the Y-conjugacy class of the element x.
This equality is classic; see for instance . Here we present a short proof: consider the bipartite graph with one side of the bipartition labeled by the elements of and the other side of the bipartition labeled by the elements of . Declare adjacent to if fixes . Clearly, has valency and has valency . Thus . ∎
The rest of the section is devoted to the proof Theorem 1.1. Let G be a transitive permutation group acting on a set Ω, , with such that the point stabilizer is a 2-group. We need to prove that . Our proof is by induction on .
Let Q be a Sylow 2-subgroup of G with , and let . Since , the set is a G-invariant partition of Σ upon which G acts with the stabilizer of the element being Q. As and Q is a 2-group, the action of G on Σ is faithful. If , then by induction for every . Therefore, from Lemma 2.1, we have for every . We may thus suppose that , that is,
Let N be a minimal normal subgroup of G, and let K be the kernel of the action of G on the set Σ of N-orbits. Suppose that N is not transitive on Ω. In particular, is a non-identity transitive permutation group on Σ of odd degree greater than 1; moreover, given and , the setwise stabilizer of B in G is , and is a Sylow 2-subgroup of . By induction, if , then , and hence . Suppose that . Since and is a Sylow 2-subgroup of G, we deduce that is a Sylow 2-subgroup of K. In particular, from the Frattini argument, we have . This implies that the core of in K equals the core of in G, which is trivial, implying that K acts faithfully on . Since this argument does not depend upon , K acts faithfully on each element of Σ. In particular, for every , the restriction of g to B is a non-identity permutation, and hence, by induction, . Since this argument does not depend upon , we have .
It remains to deal with the case that every minimal normal subgroup of G is transitive on Ω, that is, G is quasiprimitive.
The class of quasiprimitive permutation groups may be described (see ) in a fashion very similar to the description given by the O’Nan–Scott theorem for primitive permutation groups. In , this description is refined, and eight types of quasiprimitive groups are defined, namely HA, HS, HC, SD, CD, TW, PA and AS, such that every quasiprimitive group belongs to exactly one of these types. As our group G has odd degree, it is readily seen, using the terminology in [27, 28], that it is of HA, AS or PA type, that is, holomorphic abelian, almost simple or product action. We refer the reader to [27, 28] for more information on the structure of groups of HA, AS or PA type, or to  for an extensive treatment of permutation groups and Cartesian decompositions.
Assume G has O’Nan–Scott type HA; let V be the socle of G, let with , and let . As , V is an elementary abelian p-group for some prime . Then , and the action of G on Ω is permutation equivalent to the natural holomorph action of G on V, with V acting by right multiplication and with acting by conjugation. Using this identification, we have
Assume that G has O’Nan–Scott type PA. Following [27, 28], we set some notation and state some facts regarding the groups of O’Nan–Scott type PA. Let with , let , and let N be the socle of G. There exists a finite non-abelian simple group T such that for some with for each i. The group G can then be identified with a subgroup of . Let R be a Sylow 2-subgroup of T, and let Δ be the set of right cosets of R in T. As is a Sylow 2-subgroup of G, we have , where is a Sylow 2-subgroup of for each i. From [27, 28], the action of G on Ω is permutation isomorphic to the natural Cartesian product action of G on . By identifying Ω with , we have with , where W acts on Ω with the Cartesian product action. In particular, we may write the elements in the form for some and . Recall that if , then
We write .
Suppose σ is a cycle of length . Relabeling the index set , we may suppose that . From (2.2), we see that is fixed by g if and only if
Clearly, this yields that the points fixed by g are of the form
where is an arbitrary element of Δ and for each . Therefore,
Suppose now . Arguing by induction on the number of cycles in a decomposition of σ in disjoint cycles and using a computation similar to that in the previous paragraph, we obtain
As and T is a non-abelian simple group, we have and hence .
Suppose . As , without loss of generality, we may assume . Then
Since , we have and hence, by induction,
It remains to deal with the case that G is an almost simple group. Let T be the socle of G. We now divide the proof in four parts, depending on whether T is a sporadic, an alternating, an exceptional group of Lie type or a classical group.
2.1 Sporadic groups
Here we consider the situation where the socle T is one of the sporadic finite simple groups. The proof is entirely computational and uses the astonishing package “The GAP character Table Library”  implemented in the computer algebra system GAP . For sporadic groups, the proof of Theorem 1.1 follows immediately from Lemma 2.1 and from Lemma 2.3.
Let G be an almost simple primitive group on Ω with socle a sporadic simple group. Then
for every , except when in its primitive action of degree 22, where the maximum fixed-point ratio is . Moreover, for each 2-subgroup Q of and for each with , in the action of on the set of cosets of Q in , we have
the Monster and
the action of the Baby Monster on the cosets of a maximal subgroup of type ,
each permutation character of each primitive permutation representation of an almost simple group with socle a sporadic simple group is available in GAP via the package “The GAP character Table Library”. Therefore, except for the two cases mentioned above, we can quickly and easily use GAP to test the veracity of the lemma. The permutation character of the Baby Monster G on the cosets of a maximal subgroup M of type is missing from the GAP library because the conjugacy fusion of some of the elements of M in G remains a mystery; this information is vital for computing the permutation character.
In the rest of the proof, we use Lemma 2.2. Suppose then G is the Baby Monster. Let be the stabilizer in G of the point , and suppose
Using the ATLAS notation, with a direct computation, we see that
unless g is in the conjugacy class , or . In particular, the lemma is proved also in this case, except when g is in the conjugacy class or . Let us denote by t the number of solutions to the equation in . From [17, (4.6)], we see that
where and are the sets of the irreducible complex characters of of orthogonal and of symplectic type. As the character table of is available in GAP, we can compute t with this formula, and we obtain that . Therefore, when g is in the conjugacy class and , we can refine the previous bound, and we obtain
For the rest of this proof, we may assume that G is the Monster group. Let . From [30, Section 3.6], we see that the classification of the maximal subgroups of the Monster is complete except for a few small open cases. Suppose first that is isomorphic to the double cover of the Baby monster. Let π be the permutation character of G on Ω. It was proved by Breuer and Lux [4, page 2309] that
(this was also proved independently in ). With this character, we can check that no non-identity element of G fixes more than of the points. (Coincidentally, as far as we are aware, π is the only permutation character of the Monster that has been computed.) For the rest of the proof, we may assume that is not conjugate to the double cover of the Baby monster. From [30, Section 3.6], it follows that . Now, it is an easy computation to check that
except when g is in the conjugacy class and . Therefore, for the rest of the proof, we may assume that g is in the conjugacy class . From [21, Lemma 2.7], we have
This quantity can be easily computed, and it is less than . ∎
This allows us to finish the proof of Theorem 1.1 when T is a sporadic simple group. Let Σ be a maximal system of imprimitivity for G acting on Ω. Except when , the proof follows applying Lemmas 2.1 and 2.3 to Σ. When , the proof follows directly from Lemma 2.3.
2.2 Exceptional groups of Lie type
Suppose now that the socle T of G is an exceptional group of Lie type. Exactly as for the sporadic groups, we prove a much stronger statement than needed for the proof of Theorem 1.1.
Let G be an almost simple primitive group on Ω with socle an exceptional simple group of Lie type. Then for every .
For exceptional groups of Lie type, Lawther, Liebeck and Seitz [20, Theorem 1] have obtained useful and explicit upper bounds on . From these bounds, it readily follows that for every element , except when the socle of G is and . Now, and ; these cases can be analyzed with a help of a computer. The maximum fixed-point ratio for G when the socle is is (arising from the natural action of on the nine points of the projective line). The same holds for groups having socle ; the maximum is achieved on the primitive action of degree 36. ∎
2.3 Alternating groups
Suppose now that T (as an abstract group) is isomorphic to the alternating group for some with . For this proof, we argue by induction on n. We first consider the case that ; this will avoid some detour in our arguments. The result in this case follows with a computation with the invaluable computer algebra system Magma . From now on, we may assume that and hence, in particular, or . Since the action of on the cosets of one of its Sylow 2-subgroups extends to an action of , we may assume that .
Case “n odd”. Let H be a subgroup of G with and for some . (Observe that this is possible because n is odd.) Let Σ be the system of imprimitivity determined by the overgroup H of . Clearly,
As and as H acts faithfully on each of its orbits on Ω, we deduce by induction that for each . Therefore, .
Case “n even”. Let H be a subgroup of G isomorphic to the imprimitive wreath product and with for some . (Observe that this is possible because n is even.) As above,
Let , and when , let . Now, , and hence, without loss of generality, we may suppose that and . In what follows, we aim to estimate .
Now, , where P is a Sylow 2-subgroup of . The action of on the cosets of (that is, the action of G on Ω) is permutation equivalent to the natural product action of on the Cartesian product , where . (This is clear from the structure of H and , and we refer to  for more details on Cartesian decompositions of permutation groups.) Now, we can write the element g in the form or in the form , where and is the element swapping the two factors of , that is, for every . A permutation of the second kind fixes at most points, and hence
A permutation of the first kind has the property that either or . Since , we may apply induction and say that the non-identity element or fixes at most of its domain and hence so does g.
2.4 Classical groups
Suppose finally that the socle T of G is a simple classical group defined over the finite field of size q, and recall that we may assume that .
For twisted groups, our notation for q is such that and are the twisted groups contained in and , respectively. We write for some prime p and some , and we define
We let V be the natural module defined over the field of size for the covering group of T, and we let n be the dimension of V over .
In studying actions of classical groups, it is rather natural to distinguish between those actions which permute the subspaces of the natural module and those which do not. The stabilizers of subspaces are generally rather large (every parabolic subgroup falls into this class), and therefore the fixed-point ratio in these cases also tends to be rather large. As the culmination of an important series of papers [5, 6, 7, 8], Burness obtained remarkably good upper bounds on the fixed-point ratio for each finite almost simple classical group in non-subspace actions. For future reference, we first need to make the definition of non-subspace action precise.
Assume Notation 2.5. A subgroup H of G is a subspace subgroup if, for each maximal subgroup M of T that contains , one of the following conditions holds:
M is the stabilizer in T of a proper non-zero subspace U of V, where U is totally singular, or non-degenerate, or, if T is orthogonal and , a non-singular 1-subspace (U can be any subspace if );
A transitive action of G on a set Δ is a subspace action if the point stabilizer of is a subspace subgroup of G; non-subspace subgroups and actions are defined accordingly. For the convenience of the reader, we recall [5, Theorem 1].
Theorem 2.7 ([5, Theorem 1]).
Let G be a finite almost simple classical group acting transitively and faithfully on a set Δ with point stabilizer , where H is a maximal non-subspace subgroup of G. Let T be the socle of G. Then, for every , we have , where n is defined in [5, Definition 2] and either or is listed in [5, Table 1].
In Theorem 2.7, apart from and (where ), n is exactly as in Notation 2.5. The upper bound in Theorem 2.7 is quite sharp when n is large and is extremely useful for our application. However, for small values of n, Theorem 2.7 loses all of its power. For instance, when , we see from [5, Table 1] that , and hence the upper bound in Theorem 2.7 only says . However, we point out that Burness and Guest  have strengthened Theorem 2.7 for linear groups of very small rank.
We are now well equipped for the proof of Theorem 1.1 in the case where T is a non-abelian simple classical group. We use the notation that we have established above. We argue by contradiction, and we suppose that there exists with . Without loss of generality, we may assume that g is an involution. Recall that we are proving Theorem 1.1 by induction on . In particular, we may suppose that .
Case “ ”. When , [21, Theorem 1 ] yields that the pair is in [21, Tables 1 and 2]. These tables, together with the pair , have some additional information on the subgroup , on a maximal subgroup M of G containing and on . Using this detailed information, a routine case-by-case analysis yields that for every .
Case “ and ”. Since we have only a finite number of cases to check, we have proved the result invoking the help of a computer. This was done with a very naive algorithm, based on Lemma 2.1, which we now briefly explain. Given a group X and a collection of subgroup of X, we construct all maximal subgroups of Y for each . For each , we construct the permutation representation of X acting on . If X contains a non-identity permutation with , then we save M in a set ; otherwise, we disregard M from further analysis. Next, we apply this routine with replaced by . (This computation might seem very time and memory consuming, but for most groups, this procedure stops after the first iteration.) Finally, we consider the collection of all subgroups returned in the previous procedure and check that the claim of the theorem holds in each case.
For the rest of the proof, we may assume and .
Case “ , where H is a maximal non-subspace subgroup of G”. We use the result of Burness described earlier. Then and
Using the information on ι in [5, Table 1], some very basic information on the conjugacy classes of T (which can be find in  or in  for small groups) and , we found with a case-by-case analysis that (2.3) is never satisfied.
For the rest of the proof, we may assume that the only maximal subgroups H of G with are subspace subgroups. Definition 2.6 (b) does not arise here because is a Sylow 2-subgroup of G (recall (2.1)), but is even. In particular, every maximal subgroup H of G with is in the Aschbacher class .
Case “there exists a maximal subgroup H of G with , ”. Let Σ be the system of imprimitivity determined by the overgroup H of . Consider and the permutation group induced by H on Δ. Since is a Sylow 2-subgroup of G, we deduce that is a Sylow 2-subgroup of H. Therefore, the kernel of the action of H on Δ is
Thus , and hence . We have
Let . Then for some , and is the setwise stabilizer of in G. Therefore, . Since , by induction, we have that either or g fixes pointwise each element of . However, the latter case does not arise because the kernel of the action of on is , and we are assuming . Thus g does not act trivially on and . Now, (2.4) yields .
For the rest of the proof, we may suppose that every maximal subgroup H of G with is in the Aschbacher class and satisfies . In particular, , and hence H is a 2-local subgroup. Using the information on the maximal subgroups H of the finite classical groups in the Aschbacher class in [19, Section 4.1] and the fact that , we obtain and .
Case “ ”. Using again the information in [19, Section 4.1], we have that if H is a maximal subspace subgroup with and , then . In particular, we are in the position to refine slightly the argument in (2.4). Let Σ be the system of imprimitivity determined by the overgroup H of . We partition in two subsets: consists of the with g not fixing pointwise Δ, and consists of the with g fixing pointwise Δ. Observe that if , then , and hence . If contains two distinct elements Δ and , then , and hence is centralized by , where the last equality follows from the maximality of and in G and from . Therefore, . Thus, applying the inductive hypothesis for the action of g on , from (2.4), we deduce
Since , g does not act trivially on Σ, and hence . Thus
Case “ ”. Here is one of the groups , , , and .
Recall that V is the underlying module for G. Let W be a totally isotropic subspace of V of dimension 1 fixed by , and let H be the stabilizer of W. (The existence of W is guaranteed by the fact that is a 2-group and by the fact that V has characteristic 2.) Now, H is a maximal -subgroup of G with . Considering the cases we are left at this point of the proof, H is a maximal parabolic subgroup of G. Since , we deduce from the structure of H in  that g is a transvection, that is, .
Let U be the group of upper unitriangular matrices in , and let be the number of transvections in U. It is easy to show arguing inductively on n that . Using this recursive relation and the fact that , we obtain .
From the previous paragraph, , and hence
This completes the proof of Theorem 1.1.
Funding source: Javna Agencija za Raziskovalno Dejavnost RS
Award Identifier / Grant number: P1-0294
Award Identifier / Grant number: J1-1691
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