Applied differential equations, by N. Curle. Pp viii, 108. £1 paperback, £2·25 cloth. 1972 (Van Nostrand Reinhold) - 
            Finite simple groups, edited by M. B. Powell and G. Higman. Pp xi, 327. £6. 1971 (Academic Press) - 
            The theory of groups, by Homer Bechtell. Pp xii, 144. £4·90. 1971 (Addison-Wesley) - 
            Finite groups of automorphisms, by Norman Biggs. Pp iii, 117, £1·60. 1971 (Cambridge University Press) - 
            Continuous lattices, by Dana Scott. Pp iv, 37. - 
            Towards a mathematical semantics for computer languages, by Dana Scott and Christopher Strachey. Pp iv, 42. £1 each. 1971 (Oxford University Computing Laboratory Planning Research Group)

In this paper, we study the problem of how a finite group can be generated by some subgroups. In order to the finite simple groups, we show that any finite non-abelian simple group can be generated by two Sylow p1 - and p_2 -subgroups, where p_1  and p_2  are two different primes. We also show that for a given different prime numbers p  and q , any finite group can be generated by a Sylow p -subgroup and a q -subgroup.

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On the number of Sylow subgroups in finite simple groups

Journal of Algebra and Its Applications ◽

10.1142/s0219498821501152 ◽

2020 ◽

pp. 2150115

Author(s):

Zhenfeng Wu

Keyword(s):

Group Theory ◽

Simple Group ◽

Finite Groups ◽

Finite Simple Group ◽

Simple Groups ◽

Finite Simple Groups ◽

Sylow Subgroups

Denote by [Formula: see text] the number of Sylow [Formula: see text]-subgroups of [Formula: see text]. For every subgroup [Formula: see text] of [Formula: see text], it is easy to see that [Formula: see text], but [Formula: see text] does not divide [Formula: see text] in general. Following [W. Guo and E. P. Vdovin, Number of Sylow subgroups in finite groups, J. Group Theory 21(4) (2018) 695–712], we say that a group [Formula: see text] satisfies DivSyl(p) if [Formula: see text] divides [Formula: see text] for every subgroup [Formula: see text] of [Formula: see text]. In this paper, we show that “almost for every” finite simple group [Formula: see text], there exists a prime [Formula: see text] such that [Formula: see text] does not satisfy DivSyl(p).

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Variations on results on orders of products in finite groups

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2020.66 ◽

2020 ◽

pp. 1-7

Author(s):

Juan Martínez ◽

Alexander Moretó

Keyword(s):

Finite Groups ◽

Prime Power ◽

Simple Groups ◽

Prime Power Order ◽

Finite Simple Groups ◽

Prime Divisors ◽

Element Orders ◽

Hall Subgroups ◽

A Finite Group

In 2014, Baumslag and Wiegold proved that a finite group G is nilpotent if and only if o(xy) = o(x)o(y) for every x, y ∈ G with (o(x), o(y)) = 1. This has led to a number of results that characterize the nilpotence of a group (or the existence of nilpotent Hall subgroups, or the existence of normal Hall subgroups) in terms of prime divisors of element orders. Here, we look at these results with a new twist. The first of our main results asserts that G is nilpotent if and only if o(xy) ⩽ o(x)o(y) for every x, y ∈ G of prime power order with (o(x), o(y)) = 1. As an immediate consequence, we recover the Baumslag–Wiegold theorem. The proof of this result is elementary. We prove some variations of this result that depend on the classification of finite simple groups.

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On mutually permutable products of generalized supersoluble subgroups of finite groups

Asian-European Journal of Mathematics ◽

10.1142/s1793557116500546 ◽

2016 ◽

Vol 09 (03) ◽

pp. 1650054

Author(s):

E. N. Myslovets

Keyword(s):

Finite Group ◽

Simple Group ◽

Finite Groups ◽

Simple Groups ◽

Finite Simple Groups ◽

Paper Properties ◽

Supersoluble Groups ◽

A Finite Group ◽

Mutually Permutable Products ◽

Composition Factors

Let [Formula: see text] be a class of finite simple groups. We say that a finite group [Formula: see text] is a [Formula: see text]-group if all composition factors of [Formula: see text] are contained in [Formula: see text]. A group [Formula: see text] is called [Formula: see text]-supersoluble if every chief [Formula: see text]-factor of [Formula: see text] is a simple group. In this paper, properties of mutually permutable products of [Formula: see text]-supersoluble finite groups are studied. Some earlier results on mutually permutable products of [Formula: see text]-supersoluble groups (SC-groups) appear as particular cases.

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THE ‘OLD’ BRITISH HISTORIES?

The Historical Journal ◽

10.1017/s0018246x07006176 ◽

2007 ◽

Vol 50 (2) ◽

pp. 499-512 ◽

Cited By ~ 3

Author(s):

JANE H. OHLMEYER

Keyword(s):

Early Modern ◽

Early Modern Period ◽

Harvard University ◽

Academic Press ◽

Age Of Revolution ◽

Cambridge University ◽

Oxford University ◽

Modern Period ◽

Oxford University Press

Protestant war: the ‘British’ of Ireland and the wars of the three kingdoms. By Robert Armstrong. Manchester: Manchester University Press, 2005. Pp. viii+261. ISBN 0-7190-6983-1. £55.00.The origins of sectarianism in early modern Ireland. Edited by Alan Ford and John McCafferty. Cambridge: Cambridge University Press, 2005. Pp. ix+249. ISBN 0-521-83755-3. £50.00.Scottish communities abroad in the early modern period. Edited by Alexia Grosjean and Steve Murdoch. Leiden: Brill Academic Press, 2005. Pp. xxi+417. ISBN 90-04-14306-8. €147.00.Lord Broghill and the Cromwellian union with Ireland and Scotland. By Patrick Little. Woodbridge: Boydell Press, 2004. Pp. xvi+270. ISBN 184383099X. £50.00.The British revolution, 1629–1660. By Allan Macinnes. Houndsmills: Palgrave Macmillan, 2005. Pp. xi+337. ISBN 0-333-59749-4. £59.50.1659: the crisis of the Commonwealth. By Ruth E. Mayers. Woodbridge: Boydell Press, 2004. Pp. xii+306. ISBN 0861932684. £45.00.The English Atlantic in an age of revolution, 1640–1661. By Carla Gardina Pestana. Cambridge, MA: Harvard University Press, 2004. Pp. xiii+342. ISBN 0-674-01502-9. £32.95.Politics and war in the three Stuart kingdoms, 1637–1649. By David Scott. Houndsmills: Palgrave Macmillan, 2004. Pp. xiv+233. ISBN 0-333-65873-6. £52.50.The Irish and British wars, 1637–1654: triumph, tragedy, and failure. By Scott Wheeler. London: Routledge, London, 2002. Pp. x+272. ISBN 0415221315. £32.50.Britain in revolution, 1625–1660. By Austin Woolrych. Oxford: Oxford University Press, 2002. Pp. xi+814. ISBN 0-19-820081-1. £25.00.

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On the laws of certain finite groups

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700007928 ◽

1970 ◽

Vol 11 (4) ◽

pp. 441-489 ◽

Cited By ~ 8

Author(s):

John Cossey ◽

Sheila Oates MacDonald ◽

Anne Penfold Street

Keyword(s):

Finite Groups ◽

Simple Groups ◽

Finite Simple Groups ◽

Classification Problems ◽

Sylow Subgroups ◽

Local Finiteness ◽

Special Cases

In recent years a great deal of attention has been devoted to the study of finite simple groups, but one aspect which seems to have been little considered is that of the laws they satisfy. In a recent paper [3], the first two of the present authors gave a basis for laws of PSL(2, 5). The techniques of [3] can be used to show that (modulo certain classification problems) a basis for the laws of PSL(2, pn) can be made up from laws of the following types:(1) an exponent law,(2) laws which determine the Sylow subgroups,(3) laws which determine the normalisers of the Sylow subgroups,(4) in certain special cases, laws which determine subvarieties of smaller exponent, e.g. the subvariety of exponent 12 for those PSL(2, pn) which contain S4,(5) a law implying local finiteness.

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Another Existence and Uniqueness Proof of the Tits Group

Algebra Colloquium ◽

10.1142/s1005386708000242 ◽

2008 ◽

Vol 15 (02) ◽

pp. 241-278

Author(s):

Gerhard O. Michler ◽

Lizhong Wang

Keyword(s):

Finite Groups ◽

Existence And Uniqueness ◽

Frobenius Group ◽

Nilpotency Class ◽

Simple Groups ◽

Finite Simple Groups ◽

Uniqueness Proof ◽

Groups Of Lie Type ◽

Uniqueness Criteria ◽

Tits Group

In this article we give a self-contained existence and uniqueness proof for the Tits simple group T. Parrott gave the first uniqueness proof. Whereas Tits' and Parrott's results employ the theory of finite groups of Lie type, our existence and uniqueness proof follows from the general algorithms and uniqueness criteria for abstract finite simple groups described in the first author's book [11]. All we need from the previous papers is the fact that the centralizer H of the Tits group T is an extension of a 2-group J with order 29 and nilpotency class 3 by a Frobenius group F of order 20 such that the center Z(H) has order 2 and any Sylow 5-subgroup Q of H has a centralizer CJ(Q) ≤ Z(H).

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A Class of Finite Groups with Abelian Sylow p-Subgroups

Algebra Colloquium ◽

10.1142/s1005386706000411 ◽

2006 ◽

Vol 13 (03) ◽

pp. 471-480

Author(s):

Zhikai Zhang

Keyword(s):

Finite Group ◽

Finite Groups ◽

Simple Groups ◽

Finite Simple Groups ◽

Defect Groups ◽

Abelian Defect Groups ◽

A Finite Group

In this paper, we first determine the structure of the Sylow p-subgroup P of a finite group G containing no elements of order 2p (p > 2), and then show that the Broué Abelian Defect Groups Conjecture is true for the principal p-block of G. The result depends on the classification of finite simple groups.

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