scholarly journals Nilpotent residual and fitting subgroup of fixed points in finite groups

2019 ◽  
Vol 22 (6) ◽  
pp. 1059-1068
Author(s):  
Emerson de Melo
Keyword(s):  
Fixed Points ◽  
Finite Groups ◽  
Fitting Subgroup ◽  
Nilpotent Residual ◽  
Formula Group

Abstract Let q be a prime and A a finite q-group of exponent q acting by automorphisms on a finite {q^{\prime}} -group G. Assume that A has order at least {q^{3}} . We show that if {\gamma_{\infty}(C_{G}(a))} has order at most m for any {a\in A^{\#}} , then the order of {\gamma_{\infty}(G)} is bounded solely in terms of m. If the Fitting subgroup of {C_{G}(a)} has index at most m for any {a\in A^{\#}} , then the second Fitting subgroup of G has index bounded solely in terms of m.

scholarly journals FITTING SUBGROUP AND NILPOTENT RESIDUAL OF FIXED POINTS

2018 ◽  
Vol 100 (1) ◽  
pp. 61-67
Author(s):  
EMERSON DE MELO ◽  
PAVEL SHUMYATSKY
Keyword(s):  
Abelian Group ◽  
Fixed Points ◽  
Elementary Abelian Group ◽  
Fitting Subgroup ◽  
Nilpotent Residual ◽  
Formula Group

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.

scholarly journals On Finite Groups with an Automorphism of Prime Order Whose Fixed Points Have Bounded Engel Sinks

2021 ◽  
Author(s):  
E. I. Khukhro ◽  
P. Shumyatsky
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Fixed Points ◽  
Finite Groups ◽  
Prime Order ◽  
Fitting Subgroup ◽  
Engel Element ◽  
A Finite Group

AbstractA left Engel sink of an elementgof a groupGis a set$${\mathscr {E}}(g)$$E(g)such that for every$$x\in G$$x∈Gall sufficiently long commutators$$[...[[x,g],g],\dots ,g]$$[...[[x,g],g],⋯,g]belong to$${\mathscr {E}}(g)$$E(g). (Thus,gis a left Engel element precisely when we can choose$${\mathscr {E}}(g)=\{ 1\}$$E(g)={1}.) We prove that if a finite groupGadmits an automorphism$$\varphi $$φof prime order coprime to |G| such that for some positive integermevery element of the centralizer$$C_G(\varphi )$$CG(φ)has a left Engel sink of cardinality at mostm, then the index of the second Fitting subgroup$$F_2(G)$$F2(G)is bounded in terms ofm. A right Engel sink of an elementgof a groupGis a set$${\mathscr {R}}(g)$$R(g)such that for every$$x\in G$$x∈Gall sufficiently long commutators$$[\ldots [[g,x],x],\dots ,x]$$[…[[g,x],x],⋯,x]belong to$${\mathscr {R}}(g)$$R(g). (Thus,gis a right Engel element precisely when we can choose$${\mathscr {R}}(g)=\{ 1\}$$R(g)={1}.) We prove that if a finite groupGadmits an automorphism$$\varphi $$φof prime order coprime to |G| such that for some positive integermevery element of the centralizer$$C_G(\varphi )$$CG(φ)has a right Engel sink of cardinality at mostm, then the index of the Fitting subgroup$$F_1(G)$$F1(G)is bounded in terms ofm.

scholarly journals Maximal subgroups of finite groups

1990 ◽  
Vol 13 (2) ◽  
pp. 311-314
Author(s):  
S. Srinivasan
Keyword(s):  
Maximal Subgroup ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
Fitting Subgroup ◽  
Small Index

In finite groups maximal subgroups play a very important role. Results in the literature show that if the maximal subgroup has a very small index in the whole group then it influences the structure of the group itself. In this paper we study the case when the index of the maximal subgroups of the groups have a special type of relation with the Fitting subgroup of the group.

Circularity of Finite Groups without Fixed Points

2005 ◽  
Vol 144 (4) ◽  
pp. 265-273 ◽  
Author(s):  
Kostia I. Beidar ◽  
Wen-Fong Ke ◽  
Hubert Kiechle
Keyword(s):  
Fixed Points ◽  
Finite Groups

Finite groups with two supersoluble subgroups

2019 ◽  
Vol 22 (2) ◽  
pp. 297-312 ◽  
Author(s):  
Victor S. Monakhov ◽  
Alexander A. Trofimuk
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Maximal Subgroups ◽  
Sylow Subgroups ◽  
Derived Subgroup ◽  
Nilpotent Residual ◽  
A Finite Group

AbstractLetGbe a finite group. In this paper we obtain some sufficient conditions for the supersolubility ofGwith two supersoluble non-conjugate subgroupsHandKof prime index, not necessarily distinct. It is established that the supersoluble residual of such a group coincides with the nilpotent residual of the derived subgroup. We prove thatGis supersoluble in the following cases: one of the subgroupsHorKis nilpotent; the derived subgroup{G^{\prime}}ofGis nilpotent;{|G:H|=q>r=|G:K|}andHis normal inG. Also the supersolubility ofGwith two non-conjugate maximal subgroupsMandVis obtained in the following cases: all Sylow subgroups ofMand ofVare seminormal inG; all maximal subgroups ofMand ofVare seminormal inG.

scholarly journals Finite groups admitting an automorphism with two fixed points

1977 ◽  
Vol 49 (2) ◽  
pp. 547-563 ◽  
Author(s):  
Michael J Collins ◽  
Benjamin Rickman
Keyword(s):  
Fixed Points ◽  
Finite Groups

scholarly journals Engel sinks of fixed points in finite groups

2019 ◽  
Vol 223 (11) ◽  
pp. 4592-4601 ◽  
Author(s):  
Cristina Acciarri ◽  
Pavel Shumyatsky ◽  
Danilo Silveira
Keyword(s):  
Fixed Points ◽  
Finite Groups

Finite soluble groups admitting an automorphism of prime power order with few fixed points

1987 ◽  
Vol 102 (3) ◽  
pp. 431-441 ◽  
Author(s):  
Brian Hartley ◽  
Volker Turau
Keyword(s):  
Fixed Points ◽  
Soluble Group ◽  
Prime Power ◽  
Prime Power Order ◽  
Finite Soluble Group ◽  
Prime Divisors ◽  
Fitting Subgroup ◽  
Soluble Groups ◽  
Fitting Height ◽  
Number Of Prime Divisors

Let G be a finite soluble group with Fitting subgroup F(G). The Fitting series of G is defined, as usual, by F0(G) = 1 and Fi(G)/Fi−1(G) = F(G/Fi−1(G)) for i ≥ 1, and the Fitting height h = h(G) of G is the least integer such that Fn(G) = G. Suppose now that a finite soluble group A acts on G. Let k be the composition length of A, that is, the number of prime divisors (counting multiplicities) of |A|. There is a certain amount of evidence in favour of theCONJECTURE. |G:Fk(G)| is bounded by a number depending only on |A| and |CG(A)|.

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