Infinitely many finite simple groups of Lie type are not generated invariably by two elements of prime order
We show that there are infinitely many k for which Sp_{2k}(2) is not generated invariably by two elements of prime order.
math.GR↗
arXiv subjects
Robert M. Guralnick
Publications and source records attributed to Robert M. Guralnick.
Coprime automorphisms of profinite groups and the commuting probability of Sylow subgroups
Given two (closed) subgroups H, K of a profinite group G, we write Pr(H,K) for the probability that a random pair from HxK commutes. Here we are concerned with profinite groups containing Sylow subgroups P, Q such that Pr(P,Q) > 0. First, we show that if G is a profinite group containing a Sylow 2-subgroup L and a Sylow p-subgroup P , where p is odd, such that Pr(L,P) is positive, then G is virtually pro-p-soluble (Theorem 1.1). Then we handle similar issues for profinite groups admitting coprime automorphisms. In particular, we prove that if G is a profinite group admitting a group of coprime automorphisms A such that there is a Sylow 2-subgroup L of C_G(A) and an A-invariant Sylow p-subgroup P of G, where p is odd, for which Pr(L,P) > 0, then G is virtually pro-p-soluble (Theorem 1.3). On the other hand, if we only have Pr([L,A],[P,A]) > 0, then [G,A] need not be virtually pro-p-soluble. We show that in this case [G,A] has an open normal subgroup of non-p-soluble length at most 1 (Theorem 1.4). Furthermore, if P is an A-invariant Sylow p-subgroup of G such that Pr([P, A],[P, A]^x) > 0 for every x in G, then [G,A] has an open normal subgroup of non-p-soluble length at most 1 (Theorem 1.5).
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Rank properties of commutators in finite groups
For a subset $S$ of a finite group $G$, let $I_G(S)$ denote the set of commutators $[g,x]=g^{-1}g^x$, where $g\in G$ and $x\in S$. Suppose that a finite group $G$ has a Carter subgroup $C$, that is, a nilpotent subgroup containing its normalizer. Suppose that any subgroup generated by a subset of $I_G(C)$ is $r$-generated. We prove that if $G$ is soluble, then the derived subgroup $G'$ has $r$-bounded rank. We produce examples showing that the solubility condition cannot be dropped. For any finite group, we prove that the rank of $G'$ is $(r,l)$-bounded, where $l$ is the maximum rank of composition factors of $G$ isomorphic to $PSL_2(q)$ for $q\equiv 7\,(\operatorname{mod}8)$. We also prove in the general case that $G'$ has $r$-bounded rank under the additional condition that for any $x\in I_G(C)$, any subgroup generated by a subset of $I_G(x)$ is $r$-generated. The proofs rely on the classification of finite simple groups, using which we prove that if a finite simple group $G$ has an element $x$ of prime order $p$ such that any subgroup generated by a subset of $I_G(x)$ is $r$-generated, then $G$ has $r$-bounded (Prüfer) rank or is isomorphic to $PSL_2(q)$ with $q \equiv 3\,(\operatorname{mod}4)$ when $p=2$, or to $PSL_2(q)$ with $q \equiv -1\,(\operatorname{mod} p)$ when $p\ne 2$.
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Constructing a complex Lie algebra isomorphic to its complex conjugate but not definable over reals
Following an idea of Demarche and using computer calculations of the authors and ideas of the LLM Claude Fable 5, we construct an explicit 10-dimensional complex two-step nilpotent Lie algebra that is isomorphic to its complex conjugate but cannot be defined over the field of real numbers R.
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Factorizations of Almost Simple Groups with Applications
The classification of factorizations $G=HK$ of finite almost simple groups, proposed by Wielandt in 1979 and pursued through several partial classifications, has remained open in one major case. We settle that case: for every finite almost simple classical group $G$, we determine all factorizations $G=HK$ in which both $H$ and $K$ have a unique nonsolvable composition factor, completing the classification of factorizations of finite almost simple groups. Among other consequences of this classification, we prove that the smallest dimension of a biperfect bicrossproduct Hopf algebra is $41287680$.
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Forcible groups and Frattini covers
We say that a finite group $G$ *forces* a finite group $H$ if every finite cover of $G$ contains a subgroup isomorphic to $H$, and we say $H$ is *forcible* if some finite group $G$ forces $H$. Complementing a classical result of Thompson--Mann, and answering a recent question of the third author, we show that a finite group is forcible if and only if it is abelian and its Sylow subgroups are elementary-by-cyclic. We also prove relative forcibility results for abelian $p$-groups in the settings of powerful $p$-groups and $p$-groups of bounded nilpotency class.
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Rank type conditions on commutators in finite groups
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.
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Local--global generation property of commutators in finite $π$-soluble groups
For a group $A$ acting by automorphisms on a group $G$, let $I_G(A)$ denote the set of commutators $[g,a]=g^{-1}g^a$, where $g\in G$ and $a\in A$, so that $[G,A]$ is the subgroup generated by $I_G(A)$. We prove that if $A$ is a $π$-group of automorphisms of a $π$-soluble finite group $G$ such that any subset of $I_G(A)$ generates a subgroup that can be generated by $r$ elements, then the rank of $[G,A]$ is bounded in terms of $r$. Examples show that such a result does not hold without the assumption of $π$-solubility. Earlier we obtained this type of results for groups of coprime automorphisms and for Sylow $p$-subgroups of $p$-soluble groups.
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On a conjecture of Peter Neumann on fixed points in permutation groups
We prove a conjecture of Peter Neumann from 1966, predicting that every finite non-regular primitive permutation group of degree $n$ contains an element fixing at least one point and at most $n^{1/2}$ points. In fact, we prove a stronger version, where $n^{1/2}$ is replaced by $n^{1/3}$, and this is best possible. The case where $G$ is affine was proved by Guralnick and Malle; in this paper we address the case where $G$ is non-affine.
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The variety of nilpotent pairs $(A,B)$ with $[A,B] = λI$
Let $k$ be an algebraically closed field of characteristic $p >0$. We consider the variety of nilpotent pairs $(A,B)$ with $[A,B]=λI$, namely the set of pairs $ X = \{ (A,B) \in M_n(k) \times M_n(k) \mid A,B \text{ nilpotent}, [A,B]=λI, λ\in k \}$. We prove that if $n=pr$, then $X$ is irreducible of dimension $n^2$.
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Finite groups, commuting probability, and coprime automorphisms
Given two subgroups $H,K$ of a finite group $G$, the probability that a pair of random elements from $H$ and $K$ commutes is denoted by $Pr(H,K)$. Suppose that a finite group $G$ admits a group of coprime automorphisms $A$ and let $ε>0$. We show that, if for any distinct primes $p,q\inπ(G)$ there is an $A$-invariant Sylow $p$-subgroup $P$ and an $A$-invariant Sylow $q$-subgroup $Q$ of $G$ for which $Pr([P,A],[Q,A])\geε$, then $F_2([G,A])$ has $ε$-bounded index in $[G,A]$ (Theorem 1.2). Here $F_2(K)$ stands for the second term of the upper Fitting seris of a group $K$. We also show that, if $G=[G,A]$ and for any prime $p$ dividing the order of $G$ there is an $A$-invariant Sylow $p$-subgroup $P$ such that $\Pr([P,A], [P,A]^x)\geqε$ for all $x\in G$, then $G$ is bounded-by-abelian-by-bounded (Theorem 1.4).
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Invariant derivations and trace bounds
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.
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Commuting probability for conjugate subgroups of a finite group
Given two subgroups H,K of a finite group G, the probability that a pair of random elements from H and K commutes is denoted by \pr(H,K). We address the following question. Let P be a p-subgroup of a finite group G and assume that \pr(P,P^x)\geq\e>0 for every x\in G. Is the order of P modulo O_p(G) bounded in terms of e only? With respect to this question, we establish several positive results but show that in general the answer is negative. In particular, we prove that if the composition factors of G which are isomorphic to simple groups of Lie type in characteristic p, have Lie rank at most n, then the order of P modulo O_p(G) is bounded in terms of n and e only. If P is a Sylow p-subgroup of G, then the order of P modulo O_p(G) is bounded in terms e only. Some other results of similar flavour are established. We also show that if \pr(P_1,P_2)>0 for every two Sylow p-subgroups P_1,P_2 of a profinite group G, then O_{p,p'}(G) is open in G.
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On the number and sizes of double cosets of Sylow subgroups of the symmetric group
Let $P_n$ be a Sylow $p$-subgroup of the symmetric group $S_n$. We investigate the number and sizes of the $P_n\setminus S_n\ /\ P_n$ double cosets, showing that most double cosets have maximal size when $p$ is odd, or equivalently, that $P_n\cap P_n^x=1$ for most $x\in S_n$ when $n$ is large. We also find that all possible sizes of such double cosets occur, modulo a list of small exceptions.
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Fixed point ratios, Sylow numbers and coverings of $p$-elements in finite groups
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 $G$.
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Commuting Involutions in Finite Simple Groups
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.
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Invariable generation of finite simple groups and rational homology of coset posets
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.
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An extension of Gow's theorem
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.
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