For a group $G$, let $U$ be the group of units of the integral group ring $\mathbb{Z}G$. The group $G$ is said to have the normalizer property if $\text{N}_U(G)=\text{Z}(U)G$. It is shown that Blackburn groups have the normalizer property. These are the groups which have non-normal finite subgroups, with the intersection of all of them being nontrivial. Groups $G$ for which class-preserving automorphisms are inner automorphisms, $\text{Out}_c(G)=1$, have the normalizer property. Recently, Herman and Li have shown that $\text{Out}_c(G)=1$ for a finite Blackburn group $G$. We show that $\text{out}_c(G)=1$ for the members $G$ of a few classes of metabelian groups, from which the Herman--Li result follows.
Together with recent work of Hertweck, Iwaki, Jespers and Juriaans, our main result implies that, for an arbitrary group $G$, the group of hypercentral units of $U$ is contained in $\text{Z}(U)G$.
We let $φ$ be an ageometric fully irreducible outer automorphism so that its Handel-Mosher axis bundle consists of a single unique axis. We show that the centralizer $Cen(\langleφ\rangle)$ of the cyclic subgroup generated by $φ$ equals the stabilizer $\text{Stab}(Λ^+_φ)$ of the attracting lamination $Λ^+_φ$ and is isomorphic to $\mathbb Z$. We further show, via an analogous result about the commensurator, that the normalizer $N(\langleφ\rangle)$ of $\langle φ\rangle$ is isomorphic to either $\mathbb Z$ or $\mathbb Z_2 * \mathbb Z_2$.
We advance superselection theory of pure states on a $C^*$-algebra $\mathfrak{A}$ outside of the conventional (DHR) setting. First, we canonically define conjugate and second conjugate classes of such states with respect to a given reference state $ω_{\mathrm{vac}}$ and background $a \in \mathrm{Aut}(\mathfrak{A})$. Next, for some subgroups $R \varsubsetneq S \varsubsetneq G\subset \mathrm{Aut}(\mathfrak{A})$ we study the family $\{\, ω_{\mathrm{vac}}\circ s\,|\, s\in S \}$ of infrared singular states whose superselection sectors may be disjoint for different $s$. We show that their conjugate …
In this article, we study isomorphisms between simple current extensions of a simple VOA. For example, we classify the isomorphism classes of simple current extensions of some VOAs. Moreover, we consider the same simple current extension and describe the normalizer of the abelian automorphism group associated with this extension. In particular, we regard the moonshine module as simple current extensions of five subVOAs V_L^+ for 2-elementary totally even lattices L, and describe corresponding five normalizers of elementary abelian 2-group in the automorphism group of the moonshine module in te…
Let $φ\colonΓ\to G$ be a homomorphism of groups. We consider factorizations $Γ\xrightarrow{f} M\xrightarrow{g} G$ of $φ$ such that either $g$ or $f$ are universal normal maps (namely, crossed modules). These two factorizations are natural generalizations of the usual normal closure and normalizer of a subgroup. Iterating these universal factorizations yield towers related, on the one hand, to hypercentral group extensions, Bousfield's localizations, and relative Schur multipliers. Dually, they extend to a relative situation, the automorphisms tower of a centerless group. Our constructions have…
In the paper we prove that every automorphism of and elementary adjoint Chevalley group of types $A_l, D_l$, or $E_l$, over local commutative ring with 1/2 is a composition of a ring automorphism and conjugation by some matrix from the normalizer of the Chevalley group in $\GL(V)$ ($V$ is the adjoint representation space).
In this paper, we show that all Coleman automorphisms of a finite group with self-central minimal non-trivial characteristic subgroup are inner; therefore the normalizer property holds for these groups. Using our methods we show that the holomorph and wreath product of finite simple groups, among others, have no non-inner Coleman automorphisms. As a further application of our theorems, we provide partial answers to questions raised by M. Hertweck and W. Kimmerle. Furthermore, we characterize the Coleman automorphisms of extensions of a finite nilpotent group by a cyclic $p$-group. Lastly, we n…
We introduce a class of automorphisms of rooted $d$-regular trees arising from affine actions on their boundaries viewed as infinite dimensional vector spaces. This class includes, in particular, many examples of self-similar realizations of lamplighter groups. We show that for a regular binary tree this class coincides with the normalizer of the group of all spherically homogeneous automorphisms of this tree: automorphisms whose states coincide at all vertices of each level. We study in detail a nontrivial example of an automaton group that contains an index two subgroup with elements from th…
We provide 3-local characterizations of the Harada--Norton sporadic simple group and its automorphism group. Both groups are examples of groups of parabolic characteristic three and we identify them from the structure of the normalizer of the centre of a Sylow 3-subgroup.
We construct a model for the space of automorphisms of a connected p-compact group in terms of the space of automorphisms of its maximal torus normalizer and its root datum. As a consequence we show that any homomorphism to the outer automorphism group of a p-compact group can be lifted to a group action, analogous to a classical theorem of de Siebenthal for compact Lie groups. The model of this paper is used in a crucial way in our paper ``The classification of 2-compact groups'', where we prove the conjectured classification of 2-compact groups and determine their automorphism spaces.
Let 1-> H -> G _> Q -> 1 be an exact sequence of groups. In the paper of R. Oliver and J. Ventura, TAMS,362(2009), the following exact sequence was developed for centric extensions, i.e the centralizer of H in G is contained in H, 0-> H^1(Q,zH) -> Aut(G,H) -> N_{Out H}(F Q)/F Q -> H^2(Q,zH) where Aut(G,H) are the automorphisms of G which restrict to an automorphism of H, F:Q -> Out H is the outer action determined by the extension, zH is the center of H with Q-action coming from F and N_{Out H} the normalizer. It is the aim of this paper to generalize the above sequence to arbitrary extensions…
The main result of this paper describes the normalizer of a finite parabolic subgroup of a (possibly infinite) Coxeter group. We use this to compute the automorphism groups of some Lorentzian lattices and K3 surfaces.
A Mumford curve of genus g (>1) over a non-archimedean valued field k of positive characteristic has at most max{12(g-1), 2 g^(1/2) (g^(1/2)+1)^2} automorphisms. This bound is sharp in the sense that there exist Mumford curves of arbitrary high genus that attain it (they are fibre products of suitable Artin-Schreier curves). The proof provides (via its action on the Bruhat-Tits tree) a classification of discontinuous subgroups of PGL(2,k) that are normalizers of Schottky groups of Mumford curves with more than 12(g-1) automorphisms. As an application, it is shown that all automorphisms of the …
Let $\Out(F_n)$ denote the outer automorphism group of the free group $F_n$ with $n>3$. We prove that for any finite index subgroup $Γ<\Out(F_n)$, the group $\Aut(Γ)$ is isomorphic to the normalizer of $Γ$ in $\Out(F_n)$. We prove that $Γ$ is {\em co-Hopfian} : every injective homomorphism $Γ\to Γ$ is surjective. Finally, we prove that the abstract commensurator $\Comm(\Out(F_n))$ is isomorphic to $\Out(F_n)$.
Let $N \geq 2$ and let $\mathrm{Out}(F_N)$ be the outer automorphism group of a nonabelian free group of rank $N$. Let $\mathrm{IA}_N(\mathbb{Z}/3\mathbb{Z})$ be the finite index subgroup of $\mathrm{Out}(F_N)$ which is the kernel of the natural action of $\mathrm{Out}(F_N)$ on $H_1(F_N,\mathbb{Z}/3\mathbb{Z})$. We show that $\mathrm{IA}_N(\mathbb{Z}/3\mathbb{Z})$ is an $R$-group, that is, for every $ϕ,ψ\in \mathrm{IA}_N(\mathbb{Z}/3\mathbb{Z})$, if there exists $k \in \mathbb{N}^*$ such that $ϕ^k=ψ^k$, then $ϕ=ψ$. This answers a question of Handel and Mosher. We then use the fact that $\mathr…
In this paper we first show that many braid groups of low genus surfaces have their centers as direct factors. We then give a description of centralizers and normalizers of prime order elements in pure mapping class groups of surfaces with spherical quotients using automorphism groups of fundamental groups of the quotient surfaces. As an application, we use these to show that the $p$-primary part of the Farrell cohomology groups of certain mapping class groups are elementary abelian groups. At the end we compute the $p$-primary part of the Farrell cohomology of a few pure mapping class groups.
