Book contents
- Frontmatter
- Contents
- Introduction
- A speech in honour of John Cannon and Derek Holt
- Finite groups of Lie type and their representations
- Iterated monodromy groups
- Engel elements in groups
- Some classes of finite semigroups with kite-like egg-boxes of D-classes
- Structure of finite groups having few conjugacy class sizes
- Group theory in cryptography
- A survey of recent results in groups and orderings: word problems, embeddings and amalgamations
- A survey on the minimum genus and maximum order problems for bordered Klein surfaces
- On one-relator quotients of the modular group
- Miscellaneous results on supersolvable groups
- Automorphisms of products of finite groups
- A rational property of the irreducible characters of a finite group
- Automotives
- On n-abelian groups and their generalizations
- Computing with matrix groups over infinite fields
- Trends in infinite dimensional linear groups
- Engel conditions on orderable groups and in combinatorial problems (a survey)
- References
Automotives
Published online by Cambridge University Press: 05 July 2011
Book contents
- Frontmatter
- Contents
- Introduction
- A speech in honour of John Cannon and Derek Holt
- Finite groups of Lie type and their representations
- Iterated monodromy groups
- Engel elements in groups
- Some classes of finite semigroups with kite-like egg-boxes of D-classes
- Structure of finite groups having few conjugacy class sizes
- Group theory in cryptography
- A survey of recent results in groups and orderings: word problems, embeddings and amalgamations
- A survey on the minimum genus and maximum order problems for bordered Klein surfaces
- On one-relator quotients of the modular group
- Miscellaneous results on supersolvable groups
- Automorphisms of products of finite groups
- A rational property of the irreducible characters of a finite group
- Automotives
- On n-abelian groups and their generalizations
- Computing with matrix groups over infinite fields
- Trends in infinite dimensional linear groups
- Engel conditions on orderable groups and in combinatorial problems (a survey)
- References
Summary
Abstract
This paper is an eclectic collection of results of the authors. The results come from “in some sense” basic, perhaps even simple, ideas covering a variety of topics. Many of these topics are related to classic results, but others simply reflect things that were of interest to the authors.
Introduction
This is an account of some of the work done by the authors, work which falls (mostly) under the section 20D45 of the 2000 Mathematics Subject Classification. The topics are eclectic, there is no apparent connection between these “motives”, but a common trait is that the notions involved are “elementary”, or “basic,” or any other adjective suggesting simplicity.
A number of topics are related to classic results, many of which do appear in popular group theory texts. As Hardy said, “debunking” is a large part of the activity of a mathematician: trying to find simpler explanations for known results could be rewarding indeed.
Some of the themes we discuss here were visited and revisited before and we have included our results among many others for the sake of giving a larger picture. However, our approach is anything but exhaustive.
Notation and Terminology
The letter G always denotes a group. If “G is finite” is not specified, it is understood that the finiteness condition is lifted.
- Type
- Chapter
- Information
- Groups St Andrews 2009 in Bath , pp. 228 - 243Publisher: Cambridge University PressPrint publication year: 2011
References
[1] , and , Finite Groups with large automizers for their abelian subgroups, Canadian Math. Bull. (3) 40 (1997), 266–270.Google Scholar
[3] and , Finite groups with small atomizers for their nonabelian subgroups, Glasgow Math. J. 41 (1999), 59–64.Google Scholar
[4] , , and , Groups whose subgroups have small automizers, Rend. Circ. Mat. Palermo. II Ser. 48 (1999), 13–22.Google Scholar
[5] and , On the orders of automorphism groups of finite groups, Bull. London Math. Soc. 37 (2005), 381–385.Google Scholar
[6] and , On the orders of automorphism groups of finite group, II, J. Group Theory 9 (2006), 537–547.Google Scholar
[7] , On the outer automorphisms of a group, Proc. London Math. Soc. (2) 11 (1913), 40–42.Google Scholar
[8] , Theory of Groups of Finite Order, 2nd ed., Dover Publications Inc., New York, 1955.Google Scholar
[9] , and , Abelian groups as autocommutator subgroups, Arch. Math. (Basel) 90 (2008), 490–492.Google Scholar
[11] , A note on central automorphisms of groups, Atti. Accad. Naz. Lincei, Cl. Sci. Fis, Mat. Nat., IX Ser., Rend. Lincei, Mat. Appl. 3 (1992), 102–106.Google Scholar
[12] , A remark about central automorphisms of groups, Rend. Sem. Mat. Univ. Padova 115 (2006), 199–203.Google Scholar
[13] , On a special class of finite 2-groups, Glasgow Math. J. 34 (1992), no. 1, 127–131.Google Scholar
[16] and , Counting similar automorphisms of finite cyclic groups, Math. Japonica 46 (1997), 345–348.Google Scholar
[17] and , Finite groups with large atomizers for their nonabelian subgroups, Arch. Math. (Basel) 69 (1997), 441–444.Google Scholar
[18] and , Right 2-Engel elements and commuting automorphisms, J. Algebra 238 (2001), 479–484.Google Scholar
[19] , and , On commuting automorphisms of groups, Arch. Math. (Basel) 79 (2002), 423–429.Google Scholar
[20] and , On orbits of automorphism groups (Russian), Sib. Mat. Zh. 46 (2005), 533–537; English translation in Sib. Math. J.46 (2005), 413–416.Google Scholar
[22] and , On a theorem of Burnside on fixed-point-free automorphisms, Arch. Math. (Basel) 90 (2008), 97–100.Google Scholar
[24] and , On orbits of automorphism groups, II, Arch. Math. (Basel) 92 (2009), 200–205.Google Scholar
[25] , and , Forcing a finite group to be abelian, accepted by P.R.I.A.
[26] and , unpublished manuscript.
[27] , The converse of a theorem of Gashütz on Frattini subgroups, Math. Z. 224 (1997), 103–111.Google Scholar
[28] , and , Autocommutators and the autocommutator subgroup, Contemporary Math. 421 (2006), 137–146.Google Scholar
[30] , Nichtabelsche p-Gruppen besitzen äussere p-Automorphismen, J. Algebra 4 (1966), 1–2.Google Scholar
[33] and , The automorphism group of a finite p-group is almost always a p-group, J. Algebra 312 (2007), 294–329.Google Scholar
[35] , A counterexample to the isomorphism problem for integral group rings, Ann. Math. (2) 154 (2001), 115–138.Google Scholar
[37] , Risolubilitá dei gruppi finiti dotati di un automorfismo spezzante di ordine 4, Boll. Unione Mat. Ital. VII Ser. B 8 (1994), 915–928.Google Scholar
[38] , Automorfismi che fissano i centralizzanti di un gruppo, Rend. Semin. Mat. Univ. Padova 102 (1999), 233–239.Google Scholar
[39] , Automorfismi spezzanti di ordine primo, Rend. Circ. Mat. Palermo II Ser. 50 (2003), 158–162.Google Scholar
[41] and , On the co-Dedekindian finite p-groups with noncyclic abelian second centre, Glasgow Math. J. 44 (2002), 1–8.Google Scholar
[43] , On the classification of finite SANS-groups, J. Algebra Appl. 6 (2007), 461–467.Google Scholar
[46] and , On a question of Deaconescu about automorphisms, Rend. Semin. Mat. Univ. Padova 89 (1993), 83–86.Google Scholar
[48] , Proof of a conjecture by Garrett Birkoff and Philip Hall on the automorphisms of a finite group, Bull. London Math. Soc. 27 (1995), 222–224.Google Scholar
[49] , On a question of Deaconescu about automorphisms, III, Rend. Semin. Mat. Univ. Padova 99 (1998), 45–82.Google Scholar
[51] , Private communication.
[52] , Finite groups admitting a fixed-point-free automorphism group, J. Algebra 174 (1995), 724–727.Google Scholar
[53] and , On a question of Deaconescu about automorphisms, II, Rend. Semin. Mat. Univ. Padova 91 (1994), 61–64.Google Scholar
[54] , A conjugacy class as a transversal in a finite group, J. Algebra 239 (2001), 365–390.Google Scholar
[55] , n-inner automorphisms of finite groups, Proc. Amer. Math. Soc. 131 (2003), 3657–3664.Google Scholar
[56] , Finite groups with fixed-point-free automorphisms of prime order, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 578–584.Google Scholar
[57] , Finite groups with class preserving outer automorphisms, J. London Math. Soc. 22 (1947), 315–320.Google Scholar
[58] and , Solubility of finite groups admitting a coprime operator group, Boll. Unione Mat. Ital., VII, Ser. A 7 (1993), 325–331.Google Scholar
[59] , A group-theoretic proof of a theorem of Maclagan-Wedderburn, Proc. Glasgow Math. Assoc. 1 (1952), 53–63.Google Scholar