scholarly journals Some remarks on the p-homotopy type of B∑p2

1996 ◽  
Vol 38 (3) ◽  
pp. 337-342
Author(s):  
Maurizio Brunetti
Keyword(s):  
Finite Group ◽  
Homotopy Type ◽  
Sylow Subgroup ◽  
Prime Power ◽  
Homotopy Equivalent ◽  
Homotopy Equivalence ◽  
Image Position ◽  
A Finite Group

Let G be a finite group, H a copy of its p-Sylow subgroup, and N the normalizer of H in G. A theorem by Nishida [10] states the p-homotopy equivalence of suitable suspensions of BN and BG when H is abelian. Recently, in [3] the authors proved a stronger result: let ΩkH be the subgroup of H generated by elements of order pk or less; ifthen BN and BG are stably p-homotopy equivalent. The hypothesis above is obviously verified when H is abelian. In the same paper the authors recall that H does not verify such condition when p = 2 and G = SL2(Fq) for a suitable odd prime power q; in this case BG and BN are not stably 2-homotopy equivalent.

scholarly journals Finite Groups Whose Certain Subgroups of Prime Power Order Are -Semipermutable

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-8
Author(s):  
Mustafa Obaid
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Prime Power ◽  
Prime Power Order ◽  
Group A ◽  
Abelian Subgroups ◽  
A Finite Group

Let be a finite group. A subgroup of is said to be S-semipermutable in if permutes with every Sylow -subgroup of with . In this paper, we study the influence of S-permutability property of certain abelian subgroups of prime power order of a finite group on its structure.

scholarly journals Two remarks on Wall's D2 problem

2018 ◽  
Vol 167 (02) ◽  
pp. 361-368
Author(s):  
IAN HAMBLETON
Keyword(s):  
Finite Group ◽  
Homotopy Type ◽  
Homotopy Equivalent ◽  
Finite Complex ◽  
Simple Homotopy ◽  
A Finite Group

AbstractIf a finite group G is isomorphic to a subgroup of SO(3), then G has the D2-property. Let X be a finite complex satisfying Wall's D2-conditions. If π1(X) = G is finite, and χ(X) ≥ 1 − def(G), then X ∨ S2 is simple homotopy equivalent to a finite 2-complex, whose simple homotopy type depends only on G and χ(X).

SS-QUASINORMAL SUBGROUPS OF FINITE GROUPS

2010 ◽  
Vol 09 (06) ◽  
pp. 977-984 ◽  
Author(s):  
TAO ZHAO ◽  
XIANHUA LI
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Prime Power ◽  
Prime Power Order ◽  
A Finite Group

A subgroup H of a finite group G is said to be SS-quasinormal in G if there exists a supplement B of H in G such that H is permutable with every Sylow subgroup of B. In this paper, we get some new characterizations of supersolvability and p-nilpotency of G by assuming some subgroups of prime power order of G are SS-quasinormal.

Stable decompositions of classifying spaces of finite abelianp-groups

1988 ◽  
Vol 103 (3) ◽  
pp. 427-449 ◽  
Author(s):  
John C. Harris ◽  
Nicholas J. Kuhn
Keyword(s):  
Finite Group ◽  
Cohomology Theory ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Image Position ◽  
A Finite Group

LetBGbe the classifying space of a finite groupG. Consider the problem of finding astabledecompositionintoindecomposablewedge summands. Such a decomposition naturally splitsE*(BG), whereE* is any cohomology theory.

On S-Quasinormal and C-Normal Subgroups of Prime Power Order in Finite Groups

2011 ◽  
Vol 18 (04) ◽  
pp. 685-692
Author(s):  
Xuanli He ◽  
Shirong Li ◽  
Xiaochun Liu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Maximal Subgroups ◽  
Prime Power Order ◽  
Normal Subgroups ◽  
A Finite Group

Let G be a finite group, p the smallest prime dividing the order of G, and P a Sylow p-subgroup of G with the smallest generator number d. Consider a set [Formula: see text] of maximal subgroups of P such that [Formula: see text]. It is shown that if every member [Formula: see text] of is either S-quasinormally embedded or C-normal in G, then G is p-nilpotent. As its applications, some further results are obtained.

Finite Groups with Some Subgroups of Sylow Subgroups s∗-Semipermutable

2021 ◽  
Vol 58 (2) ◽  
pp. 147-156
Author(s):  
Qingjun Kong ◽  
Xiuyun Guo
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Subnormal Subgroup ◽  
Sylow Subgroups ◽  
A Finite Group

We introduce a new subgroup embedding property in a finite group called s∗-semipermutability. Suppose that G is a finite group and H is a subgroup of G. H is said to be s∗-semipermutable in G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K is s-semipermutable in G. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying 1 < |D| < |P | and study the structure of G under the assumption that every subgroup H of P with |H | = |D| is s∗-semipermutable in G. Some recent results are generalized and unified.

On the Morava K-theory of some finite 2-groups

1997 ◽  
Vol 121 (1) ◽  
pp. 7-13 ◽  
Author(s):  
BJÖRN SCHUSTER
Keyword(s):  
Finite Group ◽  
Sylow Subgroup ◽  
Cyclic Subgroup ◽  
Abelian Groups ◽  
Wreath Products ◽  
Classifying Space ◽  
Generalized Cohomology ◽  
K Theory ◽  
A Finite Group ◽  
Generalized Quaternion

For any fixed prime p and any non-negative integer n there is a 2(pn − 1)-periodic generalized cohomology theory K(n)*, the nth Morava K-theory. Let G be a finite group and BG its classifying space. For some time now it has been conjectured that K(n)*(BG) is concentrated in even dimensions. Standard transfer arguments show that a finite group enjoys this property whenever its p-Sylow subgroup does, so one is reduced to verifying the conjecture for p-groups. It is easy to see that it holds for abelian groups, and it has been proved for some non-abelian groups as well, namely groups of order p3 ([7]) and certain wreath products ([3], [2]). In this note we consider finite (non-abelian) 2-groups with maximal normal cyclic subgroup, i.e. dihedral, semidihedral, quasidihedral and generalized quaternion groups of order a power of two.

scholarly journals Automorphism orbits of finite groups

1986 ◽  
Vol 40 (2) ◽  
pp. 253-260 ◽  
Author(s):  
Thomas J. Laffey ◽  
Desmond MacHale
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Groups ◽  
Prime Power ◽  
Structure Theorem ◽  
Equivalence Classes ◽  
Prime Power Order ◽  
A Finite Group

AbstractLet G be a finite group and let Aut(G) be its automorphism group. Then G is called a k-orbit group if G has k orbits (equivalence classes) under the action of Aut(G). (For g, hG, we have g ~ h if ga = h for some Aut(G).) It is shown that if G is a k-orbit group, then kGp + 1, where p is the least prime dividing the order of G. The 3-orbit groups which are not of prime-power order are classified. It is shown that A5 is the only insoluble 4-orbit group, and a structure theorem is proved about soluble 4-orbit groups.

FINITE GROUPS WITH GIVEN NEARLY S-QUASINORMAL SUBGROUPS

2008 ◽  
Vol 01 (03) ◽  
pp. 369-382
Author(s):  
Nataliya V. Hutsko ◽  
Vladimir O. Lukyanenko ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Cyclic Subgroup ◽  
Saturated Formation ◽  
Supersoluble Groups ◽  
Sylow Subgroups ◽  
Quasinormal Subgroup ◽  
A Finite Group

Let G be a finite group and H a subgroup of G. Then H is said to be S-quasinormal in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-quasinormal in G. Then we say that H is nearly S-quasinormal in G if G has an S-quasinormal subgroup T such that HT = G and T ∩ H ≤ HsG. Our main result here is the following theorem. Let [Formula: see text] be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that [Formula: see text]. Suppose that every non-cyclic Sylow subgroup P of E has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is a non-abelian 2-group) having no supersoluble supplement in G are nearly S-quasinormal in G. Then [Formula: see text].

Finite Groups with Normal Normalizers

1968 ◽  
Vol 20 ◽  
pp. 1256-1260 ◽  
Author(s):  
C. Hobby
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
A Finite Group

We say that a finite group G has property N if the normalizer of every subgroup of G is normal in G. Such groups are nilpotent since every Sylow subgroup is normal (the normalizer of a Sylow subgroup is its own normalizer). Thus it is sufficient to study p-groups which have property N. Note that property N is inherited by subgroups and factor groups.

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