## Permutation groups Let $G\leq \textrm{Sym}(\Omega)$
be a transitive permutation group on a set $\Omega$ of size $n>1.$ A
permutation $g \in G$ is called a derangement if it has no fixed
point on $\Omega.$ A theorem of Jordan says that $G$
always contains a derangement. Notice that if $g$ is a derangement of
$G$ on $\Omega$ then so is any $G$-conjugate of $g.$ Denote by
$\Delta(G)$ the set of derangements of $G$ on $\Omega.$ Clearly,
$\Delta(G)$ is a union of $G$-conjugacy classes and let
$\kappa(G)$ be the number of conjugacy classes of derangements of $G$
on $\Omega.$ Let $\delta(G)=|\Delta(G)|/|G|$ be the proportion of
derangements in $G.$ From Jordan's theorem, we know that $\kappa(G)\ge 1.$
Derangements arose in different contexts and have many applications in probability, algebraic curves, number theory, topology, graph theory and combinatorics. The study of derangements has a long history dating back to 1708 when Montmort proved that $\delta(\textrm{S}_n)$ tends to $1/e$ when $n$ tends to infinity, where $\text{S}_n$ is the symmetric group of degree $n>1.$ This was later extended by Cameron and Cohen to all finite permutation groups $G$ of degree $n>1.$ They showed that $\delta(G)\ge 1/n$ and the equality holds if and only if $G$ is sharply $2$-transitive. Recent work of Fulman and Guralnick showed that there is an absolute constant $\epsilon>0$ such that $\delta(G)>\epsilon$ for any simple transitive permutation group $G.$ This confirms a conjecture due to Boston et al. and Shalev. Our interest in derangements comes from the investigation of zeros of characters, where we want to classify all pairs $(G,\chi)$ where $\chi\in\textrm{Irr}(G)$ is a nonlinear character of $G$ and $\chi$ has only one $G$-class of zeros. If $\chi$ is imprimitive, this problem can be reduced to the study of permutation groups with only one class of derangements. Problem 1. Classify all transitive permutation groups which have at most two classes of derangements.
In the paper: Burness, Timothy C.; Tong-Viet, Hung P. Derangements in primitive permutation groups, with an application to character theory. Q. J. Math. 66 (2015), no. 1, 63–96.
We proved that $G$ is a primitive permutation group with $\kappa(G)=1$ if and only if $G$ is sharply $2$-transitive, $G$ is an alternating group of degree $5$ or $G$ is $\textrm{PSL}_2(8).3.$ We also classified finite almost simple primitive permutation group $G$ with $\kappa(G)=2$ and showed that $\kappa(G)$ tends to infinity as $|G|$ tends to infinity. R. Guralnick recently shows that the same conclusion as above holds when $G$ is a transitive permutation group with $\kappa(G)=1.$ Another interesting topic in the study of derangements is the existence of derangements of specific order. Answering a question in Galois theory, Fein, Kantor and Schacher showed that every transitive permutation group of degree $n>1$ possesses a derangement of prime power order and the proof of this result required the classification of finite simple groups. Now in Problem 1, if $\kappa(G)=1,$ then every derangement of $G$ on $\Omega$ has the same order and by Fein-Kantor-Schacher theorem, this same order must be a prime power. This leads us to the following. Problem 2. Classify all finite transitive permutation groups whose derangements are $p$-elements for some fixed prime $p.$
Finite transitive permutation groups in which all derangements are involutions have been classified by Isaacs, Keller, Moreto and Lewis and they asked for the classification of finite permutation groups whose all derangements have order $p$ for some fixed prime $p.$ In our paper: Burness, T.C and H.P. Tong-Viet, Primitive permutation groups and derangements of prime power order, Manuscripta Math. 150 (2016), no 1-2, 255-291. we have shown that if $G$ is finite primitive permutation group such that every derangement in $G$ is a $p$-element for some fixed prime $p$, then $G$ is either almost simple or affine. Transitive groups with this property arise naturally in several different contexts. For instance, let us recall that the existence of a derangement of prime power order in any finite transitive permutation group implies that the relative Brauer group $B(L/K)$ of any finite extension $L/K$ of global fields is infinite. More precisely, if $L = K(\alpha)$ is a separable extension of $K$, then the $p$-primary component $B(L/K)_p$ is infinite if and only if the Galois group ${\rm Gal}(E/K)$ contains a derangement of $p$-power order, where $E$ is a Galois closure of $L$ over $K$. In this situation, it follows that the relative Brauer group $B(L/K)$ has a unique infinite primary component if and only if every derangement in ${\rm Gal}(E/K)$ is a $p$-element for some fixed prime $p$. Burness and I, we described all the almost simple primitive groups satisfying the hypothesis of Problem 2. To deal with affine groups, let $\textbf{F}$ be a field and let $V$ be a finite dimensional vector space over $\textbf{F}$. Let $H \leq {\rm GL}(V)$ be a finite group and let $p$ be a prime. Recall that $x \in H$ is a $p'$-element if the order of $x$ is not divisible by $p$. The pair $(H,V)$ is said to be $p'$-semiregular if every nontrivial $p'$-element of $H$ has no fixed points on $V\setminus\{0\}$. It is proved that if $G = HV \leq {\rm AGL}(V)$ is a finite affine primitive permutation group with point stabilizer $H = G_0$ and socle $V = (\mathbb{Z}_{r})^k$, where $r$ is a prime and $k \geq 1$, then every derangement in $G$ is a $p$-element for some fixed prime $p$ if and only if $r=p$ and the pair $(H,V)$ is $p'$-semiregular. Problem 2 also arose from other context. The movement $m$ of a permutation group G on a set $\Omega$ is the maximum of $|\Gamma^g\setminus\Gamma|$ for all subsets $\Gamma$ of $\Omega$ and all $g \in G.$ Suppose that $G$ is transitive on $\Omega$ with a finite movement $m,$ where $G$ is not a $2$-group and let $p$ be the least odd prime dividing $|G|.$ Then $|\Omega|\le \lfloor 2mp/(p-1)\rfloor$. Transitive permutation attaining this bound with $p$ at least 5 were classified and these groups $G$ have the property that all their derangements have the same order $p.$ Conjecture
(Mann-Praeger) If $G$ is a finite transitive permutation $p$-group for
some fixed prime $p$ and assume that all derangements of $G$ are of
order $p,$ then the exponent of $G$ is $p.$
This conjecture is known to be true when $p=2$ or $3$ by work of Mann and Praeger. Furthermore, Hassani, Khayaty, Khukhro and Praeger showed that the exponent of $G$ is bounded above by a function of $p$ only. We should mention that Isbell conjecture originated from game theory which says that if a prime $p$ is dominant in $n,$ then every finite transitive permutation group of degree $n$ has a derangement of $p$-power order, is still open. Currently, Burness and I, we also study the following problem for primitive permutation groups. Problem 3. Classify all finite transitive permutation groups whose derangements have prime power order.
With an application to zeros of characters in mind, it seems that we also need to consider a more general concept of derangements. Let $H$ be a proper subgroup of a finite group $G,$ an element $g \in G$ is called an $H$-derangement if the conjugacy class $g^G$ does not intersect $H.$ Now when the core of $H$ in $G$ is trivial then $G$ is a transitive permutation group with point stabilizer $H$ and $H$-derangements become derangements on the set of right cosets $G/H$. It is interesting to investigate Problems $1-3$ using the concept of $H$-derangements. |