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Fixed point ratios for primitive permutation groups have been extensively studied. Relying on a recent work of Burness and Guralnick, we obtain further results in the area. For a prime $p$ and a finite group $G$, we use fixed point ratios to study the number of Sylow $p$-subgroups of $G$ and the minimal size of a covering by proper subgroups of the set of $p$-elements of $G$.

We prove that if $G$ is a finite simple group and $x, y \in G$ are involutions, then $|x^G \cap C_G(y)| \rightarrow \infty$ as $|G| \rightarrow \infty$. This extends results of Guralnick-Robinson and Skresanov. We also prove a related result about $C_{G}(t)/O(C_G(t))$ that does not require the classification of finite simple groups.

If $H$ is a Hall subgroup of a finite group $G$, it was proven in 1989 using the classification of finite simple groups that all the irreducible complex characters of $H$ extend to $G$ if and only if there is $N\trianglelefteq G$ such that $HN=G$ and $H\cap N=1$. In this note we offer a modular version of this result under more general hypothesis.

For a subgroup $S$ of a group $G$, let $I_G(S)$ denote the set of commutators $[g,s]=g^{-1}g^s$, where $g\in G$ and $s\in S$, so that $[G,S]$ is the subgroup generated by $I_G(S)$. We prove that if $G$ is a $p$-soluble finite group with a Sylow $p$-subgroup $P$ such that any subgroup generated by a subset of $I_G(P)$ is $r$-generated, then $[G,P]$ has $r$-bounded rank. We produce examples showing that such a result does not hold without the assumption of $p$-solubility. Instead, we prove that if a finite group $G$ has a Sylow $p$-subgroup $P$ such that (a) any subgroup generated by a subset of $I_G(P)$ is $r$-generated, and (b) for any $x\in I_G(P)$, any subgroup generated by a subset of $I_G(x)$ is $r$-generated, then $[G,P]$ has $r$-bounded rank. We also prove that if $G$ is a finite group such that for every prime $p$ dividing $|G|$ for any Sylow $p$-subgroup $P$, any subgroup generated by a subset of $I_G(P)$ can be generated by $r$ elements, then the derived subgroup $G'$ has $r$-bounded rank. As an important tool in the proofs, we prove the following result, which is also of independent interest: if a finite group $G$ admits a group of coprime automorphisms $A$ such that any subgroup generated by a subset of $I_G(A)$ is $r$-generated, then the rank of $[G,A]$ is $r$-bounded.

We extend Gow's theorem on products of semisimple regular conjugacy classes to finite groups whose generalized Fitting subgroup is Z(G)S where S is a quasisimple group of Lie type in characteristic p and Z(G) has order prime to p., Comment: 7 pages

We prove that if G is a sufficiently large finite almost simple group of Lie type, then given a fixed nontrivial element x in G and a coset of G modulo its socle, the probability that x and a random element of the coset generate a subgroup containing the socle is uniformly bounded away from 0 (and goes to 1 if the field size goes to infinity). This is new even if G is simple. Together with results of Lucchini and Burness--Guralnick--Harper, this proves a conjecture of Lucchini and has an application to profinite groups. A key step in the proof is the determination of the limits for the proportion of elements in a classical group which fix no subspace of any bounded dimension., Comment: 26 pages

A subgroup $R$ of a finite group $G$ is weakly subnormal in $G$ if $R$ is not subnormal in $G$ but it is subnormal in every proper overgroup of $R$ in $G$. In this paper, we first classify all finite groups $G$ which contains a weakly subnormal $p$-subgroup for some prime $p$. We then determine all finite groups containing a cyclic weakly subnormal $p$-subgroup. As applications, we prove a number of variations of the Baer-Suzuki theorem using the orders of certain group elements., Comment: 25 pages; comments welcome

We show that every finite simple group is generated invariably by a Sylow subgroup and a cyclic group. It follows that that the order complex of the coset poset of an arbitrary finite group has nontrivial reduced rational homology., Comment: 14 pages; v2 has minor changes for publication

About 20 years ago, J-P.~Serre announced a bound on the trace of elements of compact Lie groups under the adjoint representation together with related results, provided indications of his proofs, and invited a better proof. This note provides a new, general method for proving such bounds; uses that method to derive Serre's bounds; gives a second proof of Serre's announced results that (we learned) closely follows his original argument; and provides lower bounds for traces of other representations of compact Lie groups and for Brauer characters of finite groups., Comment: v3 re-writes part I, expands other parts of the text, and adds an appendix on finite groups. v2 strengthens the results of part IV

In this paper, we determine the finite groups with a Sylow $r$-subgroup contained in a unique maximal subgroup. The proof involves a reduction to almost simple groups, and our main theorem extends earlier work of Aschbacher in the special case $r=2$. Several applications are presented. This includes some new results on weakly subnormal subgroups of finite groups, which can be used to study variations of the Baer-Suzuki theorem., Comment: 50 pages; to appear in Advances in Mathematics

We prove some results about closures of certain matrix varieties consisting of elements with the same centralizer dimension. This generalizes a result of Dixmier and has applications to topological generation of simple algebraic groups., Comment: 4 pages, 0 figures

It was shown in Part I that there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski-dense. Here we show that the same is true for as long as the transcendence degree of the field is at least $1$ in characteristic zero and at least $2$ in positive characteristic. We also consider related questions for surface groups.

We consider semi-continuity of certain dimensions on group schemes.

Let $G$ be a finite group, let $H$ be a core-free subgroup and let $b(G,H)$ denote the base size for the action of $G$ on $G/H$. Let $\alpha(G)$ be the number of conjugacy classes of core-free subgroups $H$ of $G$ with $b(G,H) \geqslant 3$. We say that $G$ is a strongly base-two group if $\alpha(G) \leqslant 1$, which means that almost every faithful transitive permutation representation of $G$ has base size $2$. In this paper we study the strongly base-two finite groups with trivial Frattini subgroup., Comment: 24 pages; to appear in the Vietnam Journal of Mathematics

Let $G$ be a finite group admitting a coprime automorphism $\alpha$. Let $J_G(\alpha)$ denote the set of all commutators $[x,\alpha]$, where $x$ belongs to an $\alpha$-invariant Sylow subgroup of $G$. We show that $[G,\alpha]$ is soluble or nilpotent if and only if any subgroup generated by a pair of elements of coprime orders from the set $J_G(\alpha)$ is soluble or nilpotent, respectively., Comment: final version, accepted in BLMS

Let $G$ be a finite simple group of Lie type and let $P$ be a Sylow $2$-subgroup of $G$. In this paper, we prove that for any nontrivial element $x \in G$, there exists $g \in G$ such that $G = \langle P, x^g \rangle$. By combining this result with recent work of Breuer and Guralnick, we deduce that if $G$ is a finite nonabelian simple group and $r$ is any prime divisor of $|G|$, then $G$ is generated by a Sylow $2$-subgroup and a Sylow $r$-subgroup., Comment: 17 pages; to appear in Contemporary Mathematics (Amitsur Centennial volume)

We verify that every alternating group of degree at most one quadrillion is invariably generated by an element of prime order together with an element of prime power order., Comment: 14 pages. Code available on github, see https://github.com/RussWoodroofe/invgen. Partial output in ancillary files, remainder available via Zenodo on upon request. v2 has minor changes for publication

We study the conjugacy classes of the classical affine groups. We derive generating functions for the number of classes analogous to formulas of Wall and the authors for the classical groups. We use these to get good upper bounds for the number of classes. These naturally come up as difficult cases in the study of the non-coprime k(GV) problem of Brauer., Comment: 35 pages

Let $G$ be a finite group, let $p$ be a prime and let ${\rm Pr}_p(G)$ be the probability that two random $p$-elements of $G$ commute. In this paper we prove that ${\rm Pr}_p(G) > (p^2+p-1)/p^3$ if and only if $G$ has a normal and abelian Sylow $p$-subgroup, which generalizes previous results on the widely studied commuting probability of a finite group. This bound is best possible in the sense that for each prime $p$ there are groups with ${\rm Pr}_p(G) = (p^2+p-1)/p^3$ and we classify all such groups. Our proof is based on bounding the proportion of $p$-elements in $G$ that commute with a fixed $p$-element in $G \setminus \textbf{O}_p(G)$, which in turn relies on recent work of the first two authors on fixed point ratios for finite primitive permutation groups., Comment: Revised according to referee's report. To appear in Algebra & Number Theory

Let $G$ be a finite primitive permutation group on a set $\Omega$ and recall that the fixed point ratio of an element $x \in G$, denoted ${\rm fpr}(x)$, is the proportion of points in $\Omega$ fixed by $x$. Fixed point ratios in this setting have been studied for many decades, finding a wide range of applications. In this paper, we are interested in comparing ${\rm fpr}(x)$ with the order of $x$. Our main theorem classifies the triples $(G,\Omega,x)$ as above with the property that $x$ has prime order $r$ and ${\rm fpr}(x) > 1/(r+1)$. There are several applications. Firstly, we extend earlier work of Guralnick and Magaard by determining the primitive permutation groups of degree $m$ with minimal degree at most $2m/3$. Secondly, our main result plays a key role in recent work of the authors (together with Moret\'{o} and Navarro) on the commuting probability of $p$-elements in finite groups. Finally, we use our main theorem to investigate the minimal index of a primitive permutation group, which allows us to answer a question of Bhargava., Comment: 66 pages; to appear in Adv. Math

Motivated by questions in algebraic geometry, Yifeng Huang recently derived generating functions for counting mutually annihilating matrices and mutually annihilating nilpotent matrices over a finite field. We give a different derivation of his results using statistical properties of random partitions chosen from the Cohen-Lenstra measure., Comment: 6 pages

Let $G$ be a simple algebraic group over an algebraically closed field and let $X$ be an irreducible subvariety of $G^r$ with $r \geqslant 2$. In this paper, we consider the general problem of determining if there exists a tuple $(x_1, \ldots, x_r) \in X$ such that $\langle x_1, \ldots, x_r \rangle$ is Zariski dense in $G$. We are primarily interested in the case where $X = C_1 \times \cdots \times C_r$ and each $C_i$ is a conjugacy class of $G$ comprising elements of prime order modulo the center of $G$. In this setting, our main theorem gives a complete solution to the problem when $G$ is a symplectic or orthogonal group. By combining our results with earlier work on linear and exceptional groups, this gives a complete solution for all simple algebraic groups. We also present several applications. For example, we use our main theorem to show that many faithful representations of symplectic and orthogonal groups are generically free. We also establish new asymptotic results on the probabilistic generation of finite simple groups by pairs of prime order elements, completing a line of research initiated by Liebeck and Shalev over 25 years ago., Comment: 69 pages; to appear in J. Eur. Math. Soc. (JEMS)

Let $G$ be a finite group admitting a coprime automorphism $\alpha$ of order $e$. Denote by $I_G(\alpha)$ the set of commutators $g^{-1}g^\alpha$, where $g\in G$, and by $[G,\alpha]$ the subgroup generated by $I_G(\alpha)$. We study the impact of $I_G(\alpha)$ on the structure of $[G,\alpha]$. Suppose that each subgroup generated by a subset of $I_G(\alpha)$ can be generated by at most $r$ elements. We show that the rank of $[G,\alpha]$ is $(e,r)$-bounded. Along the way, we establish several results of independent interest. In particular, we prove that if every element of $I_G(\alpha)$ has odd order, then $[G,\alpha]$ has odd order too. Further, if every pair of elements from $I_G(\alpha)$ generates a soluble, or nilpotent, subgroup, then $[G,\alpha]$ is soluble, or respectively nilpotent., Comment: Final version to appear in Transactions of the American Mathematical Society

We prove a myriad of results related to the stabilizer in an algebraic group $G$ of a generic vector in a representation $V$ of $G$ over an algebraically closed field $k$. Our results are on the level of group schemes, which carries more information than considering both the Lie algebra of $G$ and the group $G(k)$ of $k$-points. For $G$ simple and $V$ faithful and irreducible, we prove the existence of a stabilizer in general position, sometimes called a principal orbit type. We determine those $G$ and $V$ for which the stabilizer in general position is smooth, or $\dim V/G < \dim G$, or there is a $v \in V$ whose stabilizer in $G$ is trivial., Comment: Text re-organized since v2, some numbering has changed

Answering a question of Dan Haran and generalizing some results of Aschbacher-Guralnick and Suzuki, we prove that given a set of primes pi, any finite group can be generated by a pi-subgroup and a pi'-subgroup. This gives a free product description of a countably generated free profinite group., Comment: We generalized the main result in the previous version, simplified the proof and give an application to profinite groups. In the latest version, we also proved a result about generation by Sylow subgroups. The paper will appear in the Israel Journal of Mathematics

Motivated in part by representation theoretic questions, we prove that if G is a finite quasi-simple group, then there exists an elementary abelian subgroup of G that intersects every conjugacy class of involutions of G.

We prove a broad generalization of a theorem of W. Burnside on real characters using permutation characters. Under a necessary hypothesis, We can give some control on multiplicities (a result that needs the Classification of Finite Simple Groups). Along the way, we also give a new characterization of the 2-closed finite groups using odd-order real elements of the group. All this can be seen as a contribution to Brauer's Problem 11. We also obtain similar results for 2-Brauer characters. We also classify finite primitive permutation groups in which every real element has a fixed point., Comment: Minor changes. The paper will appear in Journal of Algebra

A group $G$ is said to be $\frac{3}{2}$-generated if every nontrivial element belongs to a generating pair. It is easy to see that if $G$ has this property then every proper quotient of $G$ is cyclic. In this paper we prove that the converse is true for finite groups, which settles a conjecture of Breuer, Guralnick and Kantor from 2008. In fact, we prove a much stronger result, which solves a problem posed by Brenner and Wiegold in 1975. Namely, if $G$ is a finite group and every proper quotient of $G$ is cyclic, then for any pair of nontrivial elements $x_1,x_2 \in G$, there exists $y \in G$ such that $G = \langle x_1, y \rangle = \langle x_2, y \rangle$. In other words, $s(G) \geqslant 2$, where $s(G)$ is the spread of $G$. Moreover, if $u(G)$ denotes the more restrictive uniform spread of $G$, then we can completely characterise the finite groups $G$ with $u(G) = 0$ and $u(G)=1$. To prove these results, we first establish a reduction to almost simple groups. For simple groups, the result was proved by Guralnick and Kantor in 2000 using probabilistic methods and since then the almost simple groups have been the subject of several papers. By combining our reduction theorem and this earlier work, it remains to handle the groups whose socles are exceptional groups of Lie type and this is the case we treat in this paper., Comment: 48 pages; to appear in Annals of Mathematics