Abstract
We classify the primitive but not affine permutation subgroups of \(S_n\) with the following property: There exists some unbounded subset X of size at most \(\hbox {clog}n\) whose orbit is polynomially bounded (in terms of n). This class of subgroups arises naturally in the study of threshold behavior of G-symmetric properties of Boolean functions. Our result answers a problem of Bourgain and Kalai. Contrary to a suggestion in their paper, we give an example for a primitive permutation group with this property, which contains \({A_{m}}\) as a section for \(m \rightarrow \infty \).
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1 Introduction
Let \(I = \{n_t|t \in \mathbb {N}\}\) be a sequence of the natural numbers \(\mathbb {N}\). For a sequence \(G_t \le S_{n_t}\)\((t \in \mathbb {N})\) of primitive permutation groups, we say that \(\{G_t\}\) has the \(small \,\ orbit \,\ property\) if there exist some constants b and c and some subsets \(X_t \subseteq \{1, \ldots , n_t\}\) such that \(\{ |X_t| \}\) is not bounded, for all t\(|X_t| \le \hbox {clog}n_t\), and \(|G_t(X_t)| \le {n_t}^b\).
Consider the following example:
Let G be \(A_n\) or \(S_n\) acting on \( \varOmega = \{1, \ldots , n \}\) (\(n \ge 3\)). Then, G acts transitively on the set of all subsets of \(\varOmega \) of size k for any k, and therefore, the size of the orbit of any subset X of size k is \({n \atopwithdelims ()k}\). Clearly, the \(small \,\ orbit \,\ property\) is not satisfied.
A more interesting example is the case where \(S_n\) acts on \(\varOmega ={{\{1,2,\dots ,n\}} \atopwithdelims (){2}}\) (namely all 2-subsets of \(\{1, \ldots , n \}\)), and in this case, the orbits correspond to isomorphism classes of graphs. Note that also here the \(small \,\ orbit \,\ property\) is not satisfied.
Let \(G_t\) (\(t \in \mathbb {N}\)) be a sequence of finite groups. Let \(I = \{n_t | t \in \mathbb {N} \}\). Bourgain and Kalai posed the question of understanding primitive permutation groups with the \(small \,\ orbit \,\ property\) (see [1]). They proposed that the following are equivalent for an infinite set I of natural numbers and a sequence of groups \(G_t\) (\(t \in I\)):
-
1.
For any embedding of \(G_t\) in \(S_{n_t}\) such that the action on \(\{1, \ldots , n_t\}\) is primitive, \(G_t\) satisfies the \(small \,\ orbit \,\ property\).
-
2.
There exists some M, such that \(G_t\) does not have \(A_m\) as a composition factor for m greater than M.
The motivation for studying permutation groups with the \(small \,\ orbit \,\ property\) comes from studies of threshold phenomena under symmetry and, in particular, a paper by Bourgain and Kalai [1].
Consider a monotone Boolean function \(f(x_1,x_2,\ldots ,x_n)\). Assume that f is invariant under the action of some primitive group G. Let \({\mu }_p(f)\) be the probability that \(f=1\) when every variable \(x_i\) equals 1 with probability p and equals 0 with probability \(1-p\) (independently). Unless f is a constant function, \({\mu }_p(f)\) is a strictly monotone function of p. Define the critical probability \(p_\mathrm{c}=p_\mathrm{c}(f)\) by \({\mu }_{p_\mathrm{c}}(f) = 1/2\). Let \(\epsilon > 0\) be a fixed tiny real number (say \(\epsilon = 1/1000\)) and suppose that \({\mu }_{p_1}(f) \epsilon \) and \({\mu }_{p_2}(f) = 1 - \epsilon \). The interval \([p_1,p_2]\) is called the threshold interval for the Boolean function f. Some results of Bourgain, Friedgut and Kalai show that if G satisfies the condition that the orbits of “large” subsets are “large,” then the threshold is sharp (see [2]), i.e., \(p_2-p_1=o(1)\) (as a function of n). Friedgut and Kalai [2] described an upper bound for the threshold interval for Boolean functions which is invariant under a transitive group G. Friedgut and Kalai gave an upper bound for the length of the threshold interval when the function is invariant under a transitive permutation group and showed that when \(\text {log}p_\mathrm{c}/\text {log}n \rightarrow 0\), then the length of the threshold interval is actually \(o(p_\mathrm{c}).\)
Bourgain and Kalai developed a method to show stronger results for permutation groups when there are no large sets with small orbits. They observed that no further improvement over the bounds given in [2] is possible when the group satisfies the \(small \,\ orbit \,\ property\). This leads naturally to the question considered here.
In this paper, we give a classification of all primitive but not affine groups which satisfy the \(small \,\ orbit \,\ property\). This classification shows that the two properties studied by Bourgain and Kalai are incomparable: namely, the first property of Bourgain and Kalai does not imply the second, and the second property does not imply the first.
Theorem
Let \(G_t \le S_{n_t}\) be a sequence of primitive groups. Let \(G = G_t\) for t large enough. Assuming \(G_t\) are not affine groups, the following two conditions hold: \(\{G_t\}\) satisfies the \(small \,\ orbit \,\ property\) if and only if for all sufficient large t\(G_t\) is one of the following :
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1.
\(G_t \le S_{k_t} {wr} S_{m_t}\) (product type) where \(m_t\) is not bounded.
-
2.
\(G_t \le H_t {wr} S_{m_t}\) (product type) where \(m_t\) is bounded and \(H_t\) is a primitive permutation group satisfying the \(small \,\ orbit \,\ property\).
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3.
\(G_t\) is of diagonal type.
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4.
\(G_t\) is of wreath product type.
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5.
\(G_t\) is an almost simple exceptional group of Lie type.
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6.
\(G_t\) is an almost simple classical group and any point stabilizer of \(G_t\) is an irreducible group.
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7.
\(G_t\) is an almost simple classical group and \(\mathrm{Soc}(G_t) = \hbox {PSL}(k_t,q_t), \hbox {PSp}(2k_t,q_t), \, \hbox {PU}(k_t,{q_t}^2), \,\ \hbox {PO}(2k_t+1,q_t), \,\ \hbox {PO}^+(2k_t,q_t)\) or \(\hbox {PO}^-(2k_t,q_t)\), where \(q_t\) is not bounded.
Remark
We do not have a full characterization of affine groups that satisfying the \(small \,\ orbit \,\ property\).
Remark
In our proof, we show that if \(\{G_t\}\) has the \(small \,\ orbit \,\ property\), then there exist some subset \(X_t \subseteq \{1, \ldots , n_t \}\) such that \(\{|X_t|\}\) is not bounded, \(|X_t| \le \text {log}{n_t}\) for all t and \(|G_t(X_t)| \le {n_t}^5\).
In particular, the two properties of Bourgain and Kalai are incomparable. Indeed, the following are immediate consequences of the theorem.
Corollary 1
The sequence \(S_{k_t} {wr} S_{m_t}\), where \(m_t\) is not bounded, satisfies the first property of Bourgain and Kalai, but not the second.
Corollary 2
The sequence \(\hbox {PSL}(k_t,q_t)\), where \(q_t\) is bounded satisfies the second property of Bourgain and Kalai, but not the first.
2 Proof of the theorem
Recall that by the O’Nan Scott Theorem (see [4]) every finite primitive group G is one of the following:
Let \(G_t \le S_{n_t}\) be a sequence of primitive groups. Assuming \(G_t\) is not of type (2), then \(G_t\) must be of type (1), (3), (4) or (5). Whenever \(G_t\) satisfies the \(small \,\ orbit \,\ property\), we construct a subset \(X_t\) whose size is not bounded but at most \(\text {log}n_t\), but whose orbit size is less than \({n_t}^5\). Otherwise, we give a proof that \(G_t\) does not satisfy the \(small \,\ orbit \,\ property\). Note that if \(|G_t| < {n_t}^5\), then we may take any subset \(X_t\) such that \(|X_t|\) is not bounded but of size at most \(\text {log}n_t\) and obtain \(|G_t(X_t)| \le |G_t| < {n_t}^5\). Hence, in what follows, we may assume that \(|G_t| > {n_t}^5\).
Since by enlarging the group the size of the orbits cannot go down, then whenever \(G_t\) has the \(small \,\ orbit \,\ property\), we may replace \(G_t\) by a maximal subgroup of \(S_{n_t}\) of the same type.
Case 1 Assume that \(G_t\) is an almost simple group. Then, by the classification theorem, either \(\mathrm{Soc}(G_t) = A_{k_t}\) for some \(k_t\), or \(\mathrm{Soc}(G_t)\) is a simple group of Lie type, or \(\mathrm{Soc}(G_t)\) is a sporadic simple group. By taking t large enough, we may exclude the last case. So suppose now that \(\mathrm{Soc}(G_t)\) is a Lie type group. We shall use the following.
Lemma 1
Let \(G \le S_n\) be an almost simple exceptional group of Lie type. Suppose G is primitive and n is large enough. Then, \(|G| < n^5\).
Proof
Suppose first that G is simple. Denote by m(G) the minimal index of a proper subgroup of G. Estimates for m(G) in the various cases are given in Lemma 6.4 of [6]. Comparing them to the known orders of the groups G, one easily sees that \(|G| < m(G)^{5-\epsilon }\) for some fixed \(\epsilon > 0\). Since \(m(G) \le n\), this yields \(|G| < n^{5-\epsilon }\) whenever G is an exceptional simple transitive subgroup of \(S_n\).
Now, if G is not simple, let K be its simple socle. Then, \(K \le S_n\) is transitive, so we have \(|K| \le n^{5-\epsilon }\). It is easy to see that \(|\mathrm{Out}(K)| \le \log {|K|}\) and so \(|\mathrm{Out}(K)| \le 5 \log {{n}} < n^{\epsilon }\) for large enough n. It follows that \(|G| \le |K||\mathrm{Out}(K)| < n^{5-\epsilon } n^{\epsilon } = n^5\), as required. \(\square \)
Thus, any exceptional almost simple group of Lie type satisfies the \(small \,\ orbit \,\ property\). Assume now that \(G_t\) is not exceptional. Hence, \(G_t\) is an almost simple classical group. Moreover, the values of \(|G_t|\) and \(m(G_t)\) (see [3] pages 170,175) imply that \(k_t\), the rank of \(G_t\), is at least 3. Let \(H_t \le G_t\) be a point stabilizer. Assume first that \(H_t\) is irreducible. By a theorem of M.Liebeck and J.Saxl [5], we have \(\vert G_t \vert \le {n_t}^5\), and therefore, \(G_t\) satisfies the \(small \,\ orbit \,\ property\). Assume now that \(H_t\) is a reducible group. Clearly, \(H_t\) is a maximal subgroup of \(G_t\), since \(G_t\) is primitive. The maximal reducible subgroups of the classical groups are well known, and we shall use their description in constructing the required subset \(X_t\).
Using the reducibility of the point stabilizer, it follows that there exist some \(d_t, q_t\) and \( k_t \ge 3\) such that \(\mathrm{Soc}(G_t)\) is one of the following:
Note that any sequence \(G_t\) of almost simple classical groups with the above action has the \(small \,\ orbit \,\ property\) if and only if its socle has this property. This follows from the inequality:
Therefore, we may restrict to the socle and assume \(G_t\) is simple. So let \(G_t\) be a sequence of simple classical groups with one of the actions above. We will show that if \(q_t\) is not bounded, then \(G_t\) satisfies the \(small \,\ orbit \,\ property\), and if \(q_t\) is bounded, then \(G_t\) does not satisfy the \(small \,\ orbit \,\ property\). Assume first that \(q_t\) is not bounded. Suppose first that \(G_t = PGL_{k_t}(q_t)\). Let \(\{v_{t,1}, \ldots , v_{t,k_t} \}\) be a basis for \(V_t\) and let \(H_t \le G_t\) be a point stabilizer. Then, \(H_t\) is a maximal subgroup of \(G_t\). Assume first that \(G_t\) acts on the set of all \(d_t\)-dimensional subspaces of \({F_{q_t}}^{k_t}\) for some \(d_t\) between 1 and \({k_t}/2\). Here \(n_t = \frac{({q_t}^{k_t}-1)({q_t}^{k_t}-q_t) \cdots ({q_t}^{k_t}-{q_t}^{d_t-1})}{({q_t}^{d_t}-1)({q_t}^{d_t}-q_t) \cdots ({q_t}^{d_t}-{q_t}^{d_t-1})} \sim {q_t}^{d_t(k_t-d_t)}\) and \(\text {log}n_t \sim d_t(k_t-d_t)\text {log}{q_t}\). We now construct the subset \(X_t\). For \(1 \le i \le \text {log}{q_t}\) and \(a_{t,1}, \ldots , a_{t,{\text {log}{q_t}}} \in F_{q_t}\), let
Let
Then, log\(q_t -1 \le \vert X_t \vert \le \) log\(q_t \le \) log\(n_t\), and since \(q_t\) is not bounded, \(\vert X_t \vert \) is not bounded as well. It is easy to see that any \(A_t \in \mathrm{GL}_{k_t}(q_t)\) which stabilizes the subspace \(\hbox {Sp}(v_{t,1}, \ldots , v_{t,{d_t-1}})\) and the vectors \(v_{t,d_t},v_{t,{d_t+1}}\) also stabilizes \(X_t\), and therefore,
It follows that
Thus, \(G_t\) satisfies the \(small \,\ orbit \,\ property\).
Assume now that \(G_t\) acts on the set of all pairs of subspaces \((U_t,W_t)\) such that \(U_t \oplus W_t = V_t\) and \(dim U_t = d_t\), where \(d_t \le {k_t}/2\). Then, \(n_t = {q_t}^{2d_t(k_t-d_t)}\) and log\({n_t} = 2d_t(k_t-d_t)\) log\({q_t}\). We now construct the subset \(X_t\). For \(1 \le i \le \) log\(q_t\) and \(a_{t,1}, \ldots , a_{t,{\text {log}{q_t}}} \in F_{q_t}\), let
and let
Let
It is easy to see that any \(A_t \in \mathrm{GL}_{k_t}(q_t)\) which stabilizes the subspace \(\hbox {Sp}(v_{t,1}, \ldots , v_{t,d_t-1})\), the vectors \(v_{t,d_t},v_{t,{d_t+1}}\) and the subspace \(\hbox {Sp}( v_{t,{d_t+2}}, \ldots , v_{t,k_t} )\) also stabilizes \(X_t\), and therefore,
It follows that
Thus, \(G_t\) satisfies the \(small \,\ orbit \,\ property\).
Assume now that \(G_t\) acts on the set of all pairs of subspaces \((U_t,W_t)\) such that \(U_t \oplus W_t\), \(dim U_t = d_t\), where \(d_t \le {k_t}/2\). Then, \(n_t = {q_t}^{d_t(2k_t-3d_t)}\) and log\({n_t} = d_t(2k_t-3d_t)\)log\(q_t\). We now construct the subset \(X_t\). For \(1 \le i \le \text {log}q_t\) and \(a_{t,1}, \ldots , a_{t,\text {log}{q_t}}\), let
and let
Let
It is easy to see that any \(A_t \in \mathrm{GL}_{k_t}(q_t)\) which stabilizes the subspace \(\hbox {Sp}(v_{t,1}, \ldots , v_{t,{d_{t-1}}})\), the vectors \(v_{t,d},v_{t,d_{t+1}}\) and the subspace \(\hbox {Sp}( v_{t,{d_{t+2}}}, \ldots , v_{t,k_t} )\) also stabilizes \(X_t\), and therefore,
It follows that
Thus, \(G_t\) satisfies the \(small \,\ orbit \,\ property\).
Assume now that \(G_t\) acts on the set of all pairs of subspaces \((U_t,W_t)\) such that \(U_t \subseteq W_t\), \(dim U_t = d_t\), and \(dimW_t = k_t-d_t\), where \(d_t \le {k_t}/2\). Then, \(n_t = {q_t}^{d_t(2k_t-3d_t)}\) and \( \text {log}{n_t} = d_t(2k_t-3d_t) \text {log}q_t\). We now construct the subset \(X_t\). Let \(\{v_1, \ldots , v_k\}\) be a basis for V. We now construct the subset X. For \(1 \le i \le \text {log}q_t\) and \(a_{t,1}, \ldots , a_{t,\text {log}{q_t}}\), let
and let
Let
It is easy to see that for any \(A_t \in \mathrm{GL}_{k_t}(q_t)\) which stabilizes the subspace \(\hbox {Sp}(v_{t,1}, \ldots , v_{t,{d_t-1}})\), the vectors \(v_{t,d},v_{d_t+1}\) and the subspace \(\hbox {Sp}( v_{t,{d_t+2}}, \ldots , v_{t,k_t} )\) also stabilizes \(X_t\), and therefore,
It follows that
Thus, \(G_t\) satisfies the \(small \,\ orbit \,\ property\).
Assume now that \(G_t\) is any group not of types D(c)(3) or A. Assume first that \(G_t\) acts on the set of all \(d_t\)-dimensional totally singular subspaces for some \(d_t\) between 1 and \({k_t}/2-1\). Let \(\{e_{t,1},f_{t,1}, \ldots , e_{t,{k_t}/2},f_{t,{k_t}/2} \}\) be a standard basis for \(V_t\), i.e., \((e_{t,i},e_{t,j}=(f_{t,i},f_{t,j}=1\), and \((e_{t,i}=f_{t,j}= {\delta }_{i,j}\) (it might be the case that instead of \(e_{t,k_t/2},f_{t,k_t/2}\) we will have basis elements \(x_t,y_t\), with different relations, or that the dimension is odd, and we will have an additional basis element \(x_t\), (see chapter 2 of [3]), but for our purpose it will not make a difference). For \(1 \le i \le \text {log}q_t\) and \(a_{t,1}, \ldots , a_{t,\text {log}q_t}\), let
Let
It is easy to see that any \(A_t \in G_t\) which stabilizes the subspace \(\hbox {Sp}(e_{t,d_t}, e_{t,d_t+1})\) also stabilizes \(X_t\). Using the results of chapter 2 of [3], we get that \(|G_t(X_t)| \le {n_t}^2\). Assume now that \(d_t = k_t/2\). For \(1 \le i \le \text {log}q_t\) and \(a_{t,1}, \ldots , a_{t,\text {log}q_t} \in F_{q_t}\), let
(in the symplectic case, we take \(e_{t,k_t/2}+a_{t,i}f_{t,k_t/2-1}\) instead of \(e_{t,k_t/2}-a_{t,i}f_{t,k_t/2-1}\)). Let
As before, we get that \(|G_t(X_t)| \le {n_t}^2\). Assume now that \(G_t\) acts on the set of all \(d_t\)-dimensional non-degenerate subspaces of \(V_t\). For \(1 \le i \le \text {log}q_t\) and \(a_{t,1},\ldots , a_{t,\text {log}q_t} \in F_{q_t}\), let
Let
Thus, we get that \(|G_t(X_t)| \le {n_t}^2\). Assume now that \(G_t\) is of type D(c)(3) above. Let \(\{e_{t,1},f_{t,1}, \ldots , e_{t,k_t/2},f_{t,k_t/2} \}\) be a symplectic basis for \(V_t\). For \(1 \le i \le \text {log}q_t\) and \(a_{t,1}, \ldots , a_{t,\text {log}q_t}\), let
Let
It follows that \(|G_t(X_t)| \le {n_t}^2\). Hence, whenever \(q_t\) is not bounded, \(G_t\) satisfies the \(small \,\ orbit \,\ property\). Assume now that \(q_t\) is bounded. Let \(G_t\) be any group of types \(A-D\) above. We will show that under the action of \(G_t\) on the set of all one-dimensional totally singular subspaces of \(V_t\), the size of the orbit of any unbounded subset of size at most \(c \text {log}n_t\) is not polynomial. Let \(X_t\) be any unbounded set of vectors of size at most \(c \text {log}n_t\). Let \(r_t = dim Span(X_t)\). Clearly, \(r_t\) is not bounded (since \(q_t\) is bounded). Let \(\{x_{t,1}, \ldots , x_{t,r_t} \} \subseteq X_t\) be a base for \(Span(X_t)\), and \(\{ v_{t,r_t+1}, \ldots , v_{t,k_t} \}\) its complement to a base for \(V_t\). Let \(g_t \in {(G_t)}_{X_t}\). Then, for all \( 1 \le i \le r_t\)\( g_t(x_{t,i}) \in X_t\) and \(g_t(v_{t,r_{t+i}}) \notin Span(X_t)\). Therefore,
It follows that
Since \(r_t\) is not bounded, it now follows that \(|G_t(X_t)|\) is not polynomial in \(n_t={q_t}^{k_t}\), as desired.
Case 3 Assume that \( G_t\) is of diagonal type. We will show that \(G_t\) satisfies the \(small \,\ orbit \,\ property\). We may assume that there exists some integer \(k_t\) such that \(G_t = {H_t}^{k_t}.(\mathrm{Out}(H_t) \times S_{k_t})\), where \(H_t\) is a non-abelian simple group, \(k_t\) is a positive integer and \(G_t\) acts on the set of all cosets of the diagonal group \(D_t = \{(h_t, \ldots , h_t) : h_t \in H_t \}\) Denote this set by \(\varOmega \). Here \(n_t = |{H_t}^{k_t-1}|\) and log\(n_t=(k_t-1)\text {log}|H_t|\). Since \(|G_t| = |{H_t}^{k_t}||H_t|k_t! \le {n_t}^2k_t!\), then we may assume that \(k_t\) is not bounded. Let \(L_t < H_t\) be a nontrivial subgroup of \(H_t\) and let \(1 \ne l_t \in L_t\). Let
Then, \(|X_t| = k_t \le \text {log}n_t\). Since \(|S_{k_t}(X_t)| = 1\) we have
Case 4 Assume that \(G_t\) is of product type. Let \(H_t\) be a primitive permutation group of type (1) or (3) acting on a set \({\Gamma }_t = \{1, \ldots , k_t \}\). For \(m_t > 1\), let \(W_t = H_t {wr} S_{m_t}\) and take \(W_t\) to act on \({\varOmega }_t = {{\Gamma }_t}^{m_t}\) in its natural product action. Then, \(G_t \le H_t {wr} S_{m_t}\), and \(n_t = {k_t}^{m_t}\). Assume first that \(m_t\) is not bounded. We will show that \(G_t\) satisfies the \( small \,\ orbit \,\ property\). Let \(x_t,y_t \in {\Gamma }_t\) and let
Then, \(|X_t| = m_t \le \text {log}n_t\). It is easy to see that \(|{S_{k_t}}^{m_t}(X_t)| \le {({k_t}^{m_t})}^2 = {n_t}^2\). Since \(|S_{m_t}(X_t)| = 1\), we have \(|G_t(X_t)| \le |{S_{k_t}}^{m_t}.S_{m_t}(X_t)| \le {n_t}^2|S_{m_t}(X_t)| = {n_t}^2\). Assume now that \(m_t\) is bounded. We will show that \(G_t\) has the \(small \,\ orbit \,\ property\) if and only if \(H_t\) has the \(small \,\ orbit \,\ property\): Assume first that \(H_t\) satisfies the \(small \,\ orbit \,\ property\). As we have shown (see cases 1 and 3), there exists some subset \(Y_t\) which is not bounded but of size at most log\(k_t\) such that \(|H_t(Y_t)| \le {k_t}^5\). Denote \(Y_t = \{{y^t}_1, \ldots , {y^t}_{|Y_t|}\}\). Let
Then, \(X_t\) is not bounded but of size at most log\(n_t\). Since \(|H_t(Y_t)| \le {k_t}^5\), then \(|{H_t}^{m_t}(X_t)| \le {({k_t}^5)}^{m_t} = {({k_t}^{m_t})}^5 = {n_t}^5\). Clearly, \(|S_{m_t}(X_t)| = 1\), and therefore, \(|{{H_t}^{m_t}}.S_{m_t}(X_t)| \le {n_t}^5\).
Assume now that \(H_t\) does not satisfy the \(small \,\ orbit \,\ property\). By our classification (see cases 1 and 3), \(H_t\) must be an almost simple group. A previous argument (see case 1) implies that \(\mathrm{Soc}(H_t)\) does not satisfy the \(small \,\ orbit \,\ property\) as well. We will show now that \(G_t\) does not satisfy the \(small \,\ orbit \,\ property\) as well: Let \(X_t\) be any unbounded subset of \({\varOmega }_t\) of size at most \(c \text {log}n_t\). Denote by \({X^t}_i\) the projection on the \(i'th\) axis. Since \(m_t\) is bounded, then there exists some \(1 \le i \le m_t\) such that \({X^t}_i\) is an unbounded set of size at most \(cm_t \text {log}k_t\). Hence, under our assumption \(|(\mathrm{Soc}(H_t))({X^t}_i)|\) is not polynomial in \(k_t\), and since \(m_t\) is bounded, then \(|(\mathrm{Soc}(H_t))({X^t}_i)|\) is not polynomial in \(n_t\). It now follows that \(|{(\mathrm{Soc}(H_t))}^{m_t}(X_t)|\) is not polynomial in \(n_t\). Since \({(\mathrm{Soc}(H_t))}^{m_t} \le G_t\), then we get now that \(|G_t(X_t)|\) is not polynomial in \(n_t\), as desired.
Case 5 Any group \(G_t\) of type 5 satisfies the \(small \,\ orbit \,\ property\), since any twisted wreath product group is contained in a group of product type of the form \(H_t {wr} S_{m_t}\) where \(H_t\) is of diagonal type (see [4]) .
References
Bourgain, J., Kalai, G.: Influences of variables and threshold intervals under group symmetries. Geom. Funct. Anal. 7, 438–461 (1997)
Friedgut, E., Kalai, G.: Every monotone graph property has a sharp threshold. Proc. Amer. Math. Soc. 124, 2993–3002 (1996)
Kleidman, Peter, Liebeck, Martin: The Subgroup Structure of the Finite Classical Groups, vol. 129. Cambridge University Press, Cambridge (1990)
Liebeck, M.W., Praeger, C.E., Saxl, J.: On the O’Nan–Scott theorem for finite primitive groups. J. Aust. Math. Soc. 44, 389–396 (1988)
Liebeck, M.W., Saxl, J.: Maximal subgroups of finite simple groups and their automorphism groups. Contemp. Math. 131(1), 243–259 (1992)
Liebeck, M.W., Shalev, A.: Simple groups, probabilistic methods, and a conjecture of Kantor and Lubotzky. J. Algebra 184, 31–57 (1996)
Acknowledgements
The author is grateful to Aner Shalev and Gil Kalai for their generous help and encouragement.
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Benbenisty, C. Primitive permutation groups satisfying the small orbit property and a problem of Bourgain and Kalai. J Algebr Comb 50, 225–235 (2019). https://doi.org/10.1007/s10801-018-0850-2
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DOI: https://doi.org/10.1007/s10801-018-0850-2