ON FINITE GROUPS OF EVEN ORDER WHOSE 2-SYLOW GROUP IS A QUATERNION GROUP

AbstractLet G be a group such that, for any subgroup H of G, every automorphism of H can be extended to an automorphism of G. Such a group G is said to be of injective type. The finite abelian groups of injective type are precisely the quasi-injective groups. We prove that a finite non-abelian group G of injective type has even order. If, furthermore, G is also quasi-injective, then we prove that G = K × B, with B a quasi-injective abelian group of odd order and either K = Q8 (the quaternion group of order 8) or K = Dih(A), a dihedral group on a quasi-injective abelian group A of odd order coprime with the order of B. We give a description of the supersoluble finite groups of injective type whose Sylow 2-subgroup are abelian showing that these groups are, in general, not quasi-injective. In particular, the characterisation of such groups is reduced to that of finite 2-groups that are of injective type. We give several restrictions on the latter. We also show that the alternating group A5 is of injective type but that the binary icosahedral group SL(2, 5) is not.

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Finite Groups with Nilpotent Subgroups of Even Order

Bulletin of the Iranian Mathematical Society ◽

10.1007/s41980-021-00570-2 ◽

2021 ◽

Author(s):

Yan Deng ◽

Wei Meng ◽

Jiakuan Lu

Keyword(s):

Finite Groups ◽

Even Order

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Nonabelian Sylow subgroups of finite groups of even order

Electronic Research Announcements of the American Mathematical Society ◽

10.1090/s1079-6762-98-00051-1 ◽

1998 ◽

Vol 4 (12) ◽

pp. 88-90

Author(s):

Naoki Chigira ◽

Nobuo Iiyori ◽

Hiroyoshi Yamaki

Keyword(s):

Finite Groups ◽

Sylow Subgroups ◽

Even Order

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On Finite Groups with an Abelian Sylow Group

Canadian Journal of Mathematics ◽

10.4153/cjm-1962-034-4 ◽

1962 ◽

Vol 14 ◽

pp. 436-450 ◽

Cited By ~ 22

Author(s):

Richard Brauer ◽

Henry S. Leonard

Keyword(s):

Finite Groups ◽

Irreducible Characters ◽

Collineation Groups ◽

Sylow Group

We shall consider finite groups of order of g which satisfy the following condition:(*) There exists a prime p dividing g such that if P ≠ 1 is an element of p-Sylow group ofthen the centralizer(P) of P incoincides with the centralizer() of in.This assumption is satisfied for a number of important classes of groups. It also plays a role in discussing finite collineation groups in a given number of dimensions.Of course (*) implies that is abelian. It is possible to obtain rather detailed information about the irreducible characters of groups in this class (§ 4).

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Finite Groups Which Admit An Automorphism With Few Orbits

Canadian Journal of Mathematics ◽

10.4153/cjm-1960-008-6 ◽

1960 ◽

Vol 12 ◽

pp. 73-100 ◽

Cited By ~ 2

Author(s):

Daniel Gorenstein

Keyword(s):

Abelian Group ◽

Direct Product ◽

Finite Groups ◽

Study Groups ◽

Quaternion Group ◽

Fixed Integer ◽

Fixed Element ◽

Odd Order ◽

Special Case

In the course of investigating the structure of finite groups which have a representation in the form ABA, for suitable subgroups A and B, we have been forced to study groups G which admit an automorphism ϕ such that every element of G lies in at least one of the orbits under ϕ of the elements g, gϕr(g), gϕrϕ(g)ϕ2r(g), gϕr(g)ϕr2r(g)ϕ3r(g), etc., where g is a fixed element of G and r is a fixed integer.In a previous paper on ABA-groups written jointly with I. N. Herstein (4), we have treated the special case r = 0 (in which case every element of G can be expressed in the form ϕi(gj)), and have shown that if the orders of ϕ and g are relatively prime, then G is either Abelian or the direct product of an Abelian group of odd order and the quaternion group of order 8.

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A Note on Special Local 2-Nilpotent Groups and the Solvability of Finite Groups

Algebra Colloquium ◽

10.1142/s1005386718000378 ◽

2018 ◽

Vol 25 (04) ◽

pp. 541-546

Author(s):

Jiangtao Shi ◽

Klavdija Kutnar ◽

Cui Zhang

Keyword(s):

Finite Group ◽

Nilpotent Group ◽

Finite Groups ◽

Nilpotent Groups ◽

Quaternion Group ◽

A Finite Group

A finite group G is called a special local 2-nilpotent group if G is not 2-nilpotent, the Sylow 2-subgroup P of G has a section isomorphic to the quaternion group of order 8, [Formula: see text] and NG(P) is 2-nilpotent. In this paper, it is shown that SL2(q), [Formula: see text], is a special local 2-nilpotent group if and only if [Formula: see text], and that GL2(q), [Formula: see text], is a special local 2-nilpotent group if and only if q is odd. Moreover, the solvability of finite groups is also investigated by giving two generalizations of a result from [A note on p-nilpotence and solvability of finite groups, J. Algebra 321 (2009) 1555–1560].

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On the Frobenius spectrum of a finite group

Journal of Algebra and Its Applications ◽

10.1142/s0219498817500517 ◽

2017 ◽

Vol 16 (03) ◽

pp. 1750051 ◽

Cited By ~ 1

Author(s):

Jiangtao Shi ◽

Wei Meng ◽

Cui Zhang

Keyword(s):

Finite Group ◽

Positive Integer ◽

Finite Groups ◽

Prime Divisor ◽

Complete Classification ◽

Composition Factor ◽

Frobenius Theorem ◽

Even Order ◽

A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text] any divisor of [Formula: see text], the order of [Formula: see text]. Let [Formula: see text], Frobenius’ theorem states that [Formula: see text] for some positive integer [Formula: see text]. We call [Formula: see text] a Frobenius quotient of [Formula: see text] for [Formula: see text]. Let [Formula: see text] be the set of all Frobenius quotients of [Formula: see text], we call [Formula: see text] the Frobenius spectrum of [Formula: see text]. In this paper, we give a complete classification of finite groups [Formula: see text] with [Formula: see text] for [Formula: see text] being the smallest prime divisor of [Formula: see text]. Moreover, let [Formula: see text] be a finite group of even order, [Formula: see text] the set of all Frobenius quotients of [Formula: see text] for even divisors of [Formula: see text] and [Formula: see text] the maximum Frobenius quotient in [Formula: see text], we prove that [Formula: see text] is always solvable if [Formula: see text] or [Formula: see text] and [Formula: see text] is not a composition factor of [Formula: see text].

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Class-Preserving Coleman Automorphisms of Finite Groups with Prescribed Centralizers

Algebra Colloquium ◽

10.1142/s1005386717000219 ◽

2017 ◽

Vol 24 (02) ◽

pp. 351-360

Author(s):

Zhengxing Li ◽

Hongwei Gao

Keyword(s):

Finite Group ◽

Finite Groups ◽

Inner Automorphism ◽

Coleman Automorphism ◽

Even Order ◽

A Finite Group

Let G be a finite group. It is proved that any class-preserving Coleman automorphism of G is an inner automorphism whenever G belongs to one of the following two classes of groups: (1) CN-groups, i.e., groups in which the centralizer of any element is nilpotent; (2) CIT-groups, i.e., groups of even order in which the centralizer of any involution is a 2-group. In particular, the normalizer conjecture holds for both CN-groups and CIT-groups. Additionally, some other results are also obtained.

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