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Register a new userProfinite groups and the fixed points of coprime automorphisms
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The main result of the paper is the following theorem. Let $q$ be a prime and $A$ an elementary abelian group of order $q^3$. Suppose that $A$ acts coprimely on a profinite group $G$ and assume that $C_G(a)$ is locally nilpotent for each $ain A^{#}$. Then the group $G$ is locally nilpotent.
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Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^r$ with $rgeq2$ acting on a finite $q$-group $G$. The following results are proved. We show that if all elements in $gamma_{r-1}(C_G(a))$ are $n$-Engel in $G$ for any $ain A^#$, then $gamma_{r-1}(G)$ is $k$-Engel for some ${n,q,r}$-bounded number $k$, and if, for some integer $d$ such that $2^dleq r-1$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $G$ for any $ain A^#$, then the $d$th derived group $G^{(d)}$ is $k$-Engel for some ${n,q,r}$-bounded number $k$. Assuming $rgeq 3$ we prove that if all elements in $gamma_{r-2}(C_G(a))$ are $n$-Engel in $C_G(a)$ for any $ain A^#$, then $gamma_{r-2}(G)$ is $k$-Engel for some ${n,q,r}$-bounded number $k$, and if, for some integer $d$ such that $2^dleq r-2$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $C_G(a)$ for any $ain A^#,$ then the $d$th derived group $G^{(d)}$ is $k$-Engel for some ${n,q,r}$-bounded number $k$. Analogue (non-quantitative) results for profinite groups are also obtained.
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.
If K is a discrete group and Z is a K-spectrum, then the homotopy fixed point spectrum Z^{hK} is Map_*(EK_+, Z)^K, the fixed points of a familiar expression. Similarly, if G is a profinite group and X is a discrete G-spectrum, then X^{hG} is often given by (H_{G,X})^G, where H_{G,X} is a certain explicit construction given by a homotopy limit in the category of discrete G-spectra. Thus, in each of two common equivariant settings, the homotopy fixed point spectrum is equal to the fixed points of an explicit object in the ambient equivariant category. We enrich this pattern by proving in a precise sense that the discrete G-spectrum H_{G,X} is just a profinite version of Map_*(EK_+, Z): at each stage of its construction, H_{G,X} replicates in the setting of discrete G-spectra the corresponding stage in the formation of Map_*(EK_+, Z) (up to a certain natural identification).
Let G be a finitely generated relatively hyperbolic group. We show that if no peripheral subgroup of G is hyperbolic relative to a collection of proper subgroups, then the fixed subgroup of every automorphism of G is relatively quasiconvex. It follows that the fixed subgroup is itself relatively hyperbolic with respect to a natural family of peripheral subgroups. If all peripheral subgroups of G are slender (respectively, slender and coherent), our result implies that the fixed subgroup of every automorphism of G is finitely generated (respectively, finitely presented). In particular, this happens when G is a limit group, and thus for any automorphism phi of G, Fix(phi) is a limit subgroup of G.
We introduce and investigate a class of profinite groups defined via extensions of centralizers analogous to the extensively studied class of finitely generated fully residually free groups, that is, limit groups (in the sense of Z. Sela). From the fact that the profinite completion of limit groups belong to this class, results on their group-theoretical structure and homological properties are obtained.
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