Let $q$ be a prime and let $A$ be an elementary abelian group of order at least $q^{3}$ acting by automorphisms on a finite $q^{\prime }$-group $G$. We prove that if $|\unicode[STIX]{x1D6FE}_{\infty }(C_{G}(a))|\leq m$ for any $a\in A^{\#}$, then the order of $\unicode[STIX]{x1D6FE}_{\infty }(G)$ is $m$-bounded. If $F(C_{G}(a))$ has index at most $m$ in $C_{G}(a)$ for any $a\in A^{\#}$, then the index of $F_{2}(G)$ is $m$-bounded.

In this paper, we study the integer sequence $(E_{n})_{n\geq 1}$ , where $E_{n}$ counts the number of possible dimensions for centralisers of $n\times n$ matrices. We give an example to show another combinatorial interpretation of $E_{n}$ and present an implicit recurrence formula for $E_{n}$ , which may provide a fast algorithm for computing $E_{n}$ . Based on the recurrence, we obtain the asymptotic formula $E_{n}=\frac{1}{2}n^{2}-\frac{2}{3}\sqrt{2}n^{3/2}+O(n^{5/4})$ .

We show that the assignment of the (left) completely bounded multiplier algebra $M_{cb}^{l}({{L}^{1}}(\mathbb{G}))$ to a locally compact quantum group $\mathbb{G}$, and the assignment of the intrinsic group, form functors between appropriate categories. Morphisms of locally compact quantum groups can be described by Hopf $*$-homomorphisms between universal ${{C}^{*}}$-algebras, by bicharacters, or by special sorts of coactions. We show that the whole theory of completely bounded multipliers can be lifted to the universal ${{C}^{*}}$-algebra level, and that the different pictures of both multipliers (reduced, universal, and as centralisers) and morphisms then interact in extremely natural ways. The intrinsic group of a quantum group can be realised as a class of multipliers, and so our techniques immediately apply. We also show how to think of the intrinsic group using the universal ${{C}^{*}}$-algebra picture, and then, again, how the differing views on the intrinsic group interact naturally with morphisms. We show that the intrinsic group is the “maximal classical” quantum subgroup of a locally compact quantum group, that it is even closed in the strong Vaes sense, and that the intrinsic group functor is an adjoint to the inclusion functor from locally compact groups to quantum groups.

Let 𝕂 be a field, char(𝕂)≠2, and G a subgroup of GL(n,𝕂). Suppose g↦g♯ is a 𝕂-linear antiautomorphism of G, and then define G1={g∈G∣g♯g=I}. For C being the centraliser 𝒞G (G1) , or any subgroup of the centre 𝒵(G) , define G(C) ={g∈G∣g♯g∈C}. We show that G(C) is a subgroup of G, and study its structure. When C=𝒞G (G1) , we have that G(C) =𝒩G (G1) , the normaliser of G1 in G. Suppose 𝕂 is algebraically closed, 𝒞G (G1) consists of scalar matrices and G1 is a connected subgroup of an affine group G. Under the latter assumptions, 𝒩G (G1) is a self-normalising subgroup of G. This holds for a number of interesting pairs (G,G1); in particular, for those that we call parabolic pairs. As well, for a certain specific setting we generalise a standard result about centres of Borel subgroups.