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A rigid automorphism of a linking system is an automorphism which restricts to the identity on the Sylow subgroup. A rigid inner automorphism is conjugation by an element in the center of the Sylow subgroup. At odd primes, it is known that each rigid automorphism of a centric linking system is inner. We prove that the group of rigid outer automorphisms of a linking system at the prime $2$ is elementary abelian, and that it splits over the subgroup of rigid inner automorphisms. In a second result, we show that if an automorphism of a finite group $G$ restricts to the identity on the centric linking system for $G$, then it is of $p$-order modulo the group of inner automorphisms, provided $G$ has no nontrivial normal $p$-subgroups. We present two applications of this last result, one to tame fusion systems.
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A. Chermak has recently proved that to each saturated fusion system over a finite $p$-group, there is a unique associated centric linking system. B. Oliver extended Chermaks proof by showing that all the higher cohomological obstruction groups relevant to unique existence of centric linking systems vanish. Both proofs indirectly assume the classification of finite simple groups. We show how to remove this assumption, thereby giving a classification-free proof of the Martino-Priddy conjecture concerning the $p$-completed classifying spaces of finite groups. Our main tool is a 1971 result of the first author on control of fixed points by $p$-local subgroups. This result is directly applicable for odd primes, and we show how a slight variation of it allows applications for $p=2$ in the presence of offenders.
An odd Coxeter group $W$ is one which admits a Coxeter system $(W,S)$ for which all the exponents $m_{ij}$ are either odd or infinity. The paper investigates the family of odd Coxeter groups whose associated labeled graphs $mathcal{V}_{(W,S)}$ are trees. It is known that two Coxeter groups in this family are isomorphic if and only if they admit Coxeter systems having the same rank and the same multiset of finite exponents. In particular, each group in this family is isomorphic to a group that admits a Coxeter system whose associated labeled graph is a star shaped tree. We give the complete description of the automorphism group of this group, and derive a sufficient condition for the splitting of the automorphism group as a semi-direct product of the inner and the outer automorphism groups. As applications, we prove that Coxeter groups in this family satisfy the $R_infty$-property and are (co)-Hopfian. We compare structural properties, automorphism groups, $R_infty$-property and (co)-Hopfianity of a special odd Coxeter group whose only finite exponent is three with the braid group and the twin group.
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 $gin G$, 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.
Suppose that p is an odd prime and G is a finite group having no normal non-trivial p-subgroup. We show that if a is an automorphism of G of p-power order centralizing a Sylow p-group of G, then a is inner. This answers a conjecture of Gross. An easy corollary is that if p is an odd prime and P is a Sylow p-subgroup of G, then the center of P is contained in the generalized Fitting subgroup of G. We give two proofs both requiring the classification of finite simple groups. For p=2, the result fails but Glauberman in 1968 proved that the square of a is inner. This answered a problem of Kourovka posed in 1999.
Let $Gamma_d(q)$ denote the group whose Cayley graph with respect to a particular generating set is the Diestel-Leader graph $DL_d(q)$, as described by Bartholdi, Neuhauser and Woess. We compute both $Aut(Gamma_d(q))$ and $Out(Gamma_d(q))$ for $d geq 2$, and apply our results to count twisted conjugacy classes in these groups when $d geq 3$. Specifically, we show that when $d geq 3$, the groups $Gamma_d(q)$ have property $R_{infty}$, that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when $d=2$ the lamplighter groups $Gamma_2(q)=L_q = {mathbb Z}_q wr {mathbb Z}$ have property $R_{infty}$ if and only if $(q,6) eq 1$.
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