## Contact Information

Professor Emeritus

Department of Mathematics

University of Illinois

1409 W.Green Street

Urbana, IL 61801

Department of Mathematics

University of Illinois

1409 W.Green Street

Urbana, IL 61801

## Research Interests

Algebra. Theory of groups, in particular infinite soluble groups. Connections between group theory, ring theory and homological algebra. Finiteness conditions. Algorithms for groups. Permutability of subgroups. Graphs of groups.

## Education

- B.Sc. Edinburgh University 1960.
- Ph.D. Cambridge University 1963.

## Awards and Honors

- Napier Medal. 1960.
- Sir Edmund Whittaker Prize 1970.
- Alexander von Humboldt Senior Scientist Prize 1979.
- Associate, Center for Advanced Study, Urbana, IL, 1981-82, 1992.
- EPSRC Visiting Fellow, Queen Mary College, 1984.
- EPSRC Visiting Fellow, Warwick University, 1999.
- Beckman Research Award 1994.
- W.W. Boone Distinguished Research Professor 2003-5.

## Academic appointments

- 1963--65 Instructor, University of Illinois.
- 1965--68 Lecturer, Queen Mary College, University of London.
- 1968--69 Assistant Professor, University of Illinois.
- 1969--74 Associate Professor, University of Illinois.
- 1974--2007 Professor, University of Illinois.
- 2007 Emeritus Professor, University of Illinois.

## Visiting academic appointments

- ETH- Zurich, 1973-4 .
- University of Florence, 1976.
- University of Freiburg, 1979-80.
- University of Wurzburg, 1982-84.
- University of Naples, 1983, 2004.
- University of Padua, 1983.
- National University of Singapore, 1990-1.
- University of Salerno, 2011, 2016.

## PhD Students

- C. D. H. Cooper
- D. McDougal
- T. A. Fournelle
- J. Zimmerman
- R.A. Nehmer
- R. D. Blyth
- V. Walter
- Y. F. Wu
- D. H. Paek
- C. C. Ferguson
- J. W. Elliot

## Editorships

- Editor: Ricerche di Matematica, International Journal of Group Theory.
- Former editor: Communications in Algebra, Journal of Group Theory.

## Books

- Finiteness Conditions and Generalized Soluble Groups, Springer, Berlin, 1972.
- A Course in Linear Algebra with Applications, 2nd ed., World Scientific, Singapore 2006.
- A Course in the Theory of Groups, 2nd ed., Springer, New York, 1995.
- Abstract Algebra. An Introduction with applications, 3rd ed., De Gruyter, Berlin, 2022.
- The Theory of Infinite Soluble Groups, (with J.C. Lennox), Oxford, 2004.
- Algebraic Models for Accounting Systems, (with S. Cruz Rambaud, J. Garcia Perez and R.A. Nehmer), World Scientific, Singapore, 2010.

## Book edited

- The Mathematical Legacy of Reinhold Baer, with (P. Griffith), Illinois J. Math., Urbana, IL 2003.

### Publications

- Groups in which normality is a transitive relation, Proc. Cambridge Philos. Soc. 60 (1964), 21-38.
- Joins of subnormal subgroups, Illinois J. Math. 9 (1965), 144-168.
- On finitely generated soluble groups, Proc. London Math. Soc.(3) 15 (1965), 508-516.
- On the theory of subnormal subgroups, Math. Z. 89 (1965), 30-51.
- Wreath products and indices of subnormality, Proc. London Math. Soc. (3)17 (1967), 257-270.
- On soluble minimax groups, Math. Z. 101 (1967), 13-40.
- Infinite Soluble and Nilpotent Groups, Queen Mary College Mathematics Notes, London (1967).
- Residual properties of some classes of infinite soluble groups, Proc. London Math. Soc. (3) 18 (1968), 495-520.
- A note on finite groups in which normality is transitive, Proc. Amer. Math. Soc. 19 (1968), 933-937.
- Finiteness conditions for subnormal and ascendant abelian subgroups, J. Algebra 10 (1968), 333-359.
- A property of the lower central series of a group, Math. Z. 107 (1968), 225-231.
- A note on groups of finite rank, Compositio Math. 21 (1969), 240-246.
- Groups which are minimal with respect to normality being intransitive, Pacific J. Math. 31 (1969), 777-785.
- A theorem on finitely generated hyperabelian groups, Invent. Math.10 (1970), 38-43.
- On the theory of groups with extremal layers, J. Algebra 14 (1970), 182-193.
- Maximal subgroups and chief factors of certain generalized soluble groups, (with R.E. Phillips and J.E. Roseblade), Pacific J. Math. 37 (1971), 475-480.
- Intersections of primary powers of a group, Math. Z. 124 (1972), 119-132.
- Groups whose images have a transitive normality relation, Conference on Group Theory, University of Wisconsin-Parkside, Lecture Notes in Math., vol. 319, 148-155, Springer, Berlin (1973).
- Groups whose homomorphic images have a transitive normality relation, Trans. Amer. Math. Soc. 196 (1973), 181-213.
- The subnormal coalescence of some classes of groups of finite rank, (with M. Drukker and I.N. Stewart), J. Austral. Math. Soc. 16 (1973), 324-327.
- Hypercentral ideals, noetherian modules and a theorem of Stroud, J. Algebra 32 (1974), 234-239.
- On the cohomology of soluble groups of finite rank, J. Pure Appl. Algebra 6 (1975), 234-239.
- Splitting theorems for infinite groups, Symposia Math. 17 (1976), 441-470.
- A new treatment of soluble groups with finiteness conditions on their abelian subgroups, Bull. London Math. Soc. 8 (1976), 113-129: Russian transl. Uspekhi Mat. Nauk. 34 (1979), 197-215 .
- The vanishing of certain homology and cohomology groups, J. Pure Appl. Algebra 7 (1976), 145-167.
- A contribution to the theory of groups with finitely many automorphisms, Proc. London Math. Soc. (3)35 (1977), 34-54.
- Homology of group extensions whose kernels are divisible abelian torsion groups, J. Pure Appl. Algebra 14 (1979), 145-165.
- Infinite torsion groups as automorphism groups, Quarterly J. Math. 30(1979), 351-364.
- On the homology of hypercentral groups, Arch. Math. (Basel) 32 (1979), 224-226.
- Finite metabelian groups with no outer automorphisms, (with T. M. Gagen), Arch. Math. (Basel) 32 (1979), 417-423.
- Soluble products of nilpotent groups, (with J.C. Lennox), Rend. Sem. Mat. Univ. Padova 62 (1980), 261-280.
- Outer automorphisms of infinite soluble groups, Rend. Sem. Mat. Univ. Padova 62 (1980), 281-294.
- On finite complete groups, (with B. Hartley), Arch. Math. (Basel) 35 (1980), 67-74.
- Recent results on finite complete groups, Algebra-Carbondale (1980), pp.178-185, Lecture Notes in Math., vol. 848, Springer-Verlag, New York, 1981.
- Nearly maximal subgroups of finitely generated soluble groups, (with J. C. Lennox), Arch. Math. (Basel) 38 (1982), 289-295.
- Groups with prescribed automorphism group, Proc. Edinburgh Math. Soc. 25 (1982), 217-227.
- Some finitely presented soluble groups which are not nilpotent by abelian by finite, (with R. Strebel), J. London Math. Soc. (2)26 (1982), 435-440.
- Applications of cohomology to the theory of groups, Groups-St. Andrews 1981, pp. 46-80, London Math. Lecture Notes, vol. 71 (1982). Reprinted 2008.
- Automorphism groups of FC-groups, (with S.E.Stonehewer and J. Wiegold), Arch. Math. (Basel) 40 (1983), 401-404.
- Infinite factorized groups, Rend. Sem. Mat-Fis. Milan 53 (1983), 347-355.
- Groups with prescribed automorphism group: a clarification, Proc. Edinburgh Math. Soc. 27 (1984), 59-60.
- The word problem for finitely generated soluble groups of finite rank, (with F.B. Cannonito), Bull. London Math. Soc. 16 (1984), 43-46.
- Soluble groups with many polycyclic quotients, (with J.S. Wilson), Proc. London Math. Soc. (3)48 (1984), 193-229.
- Soluble groups which are products of nilpotent minimax groups, (with B. Amberg), Arch. Math. (Basel) 42 (1984), 385-390.
- Topics in the theory of finitely generated soluble groups, Lecture Notes Series, National University of Singapore 21 (1984).
- Groups with boundedly finite automorphism classes, (with J. Wiegold), Rend. Sem. Mat. Univ. Padova 71 (1984), 273-286.
- Decision problems for infinite soluble groups, Groups-Korea 1983, pp. 111-117, Lecture Notes in Mathematics, vol. 1098, Springer-Verlag, Berlin, 1984.
- Automorphism groups of group extensions, Algebra and its Applications, New Delhi 1981, pp.163-167, Lecture Notes in Pure and Applied Mathematics, vol. 91, Dekker, New York, 1984.
- Finiteness, solubility and nilpotence, Essays for Philip Hall, pp. 159-206 Academic Press, New York-London, 1984.
- A permutational property of groups, (with M. Curzio, P. Longobardi and M. Maj), Arch. Math. (Basel) 44 (1985), 385-389.
- Soluble products of nilpotent groups, J. Algebra 98 (1986), 183-196.
- Decision problems for soluble groups of finite rank, Illinois J. Math. 30 (1986), 197-213.
- On torsion in groups whose automorphism groups have finite rank, (with S. Franciosi and F. de Giovanni), Rocky Mountain J. Math. 17 (1987), 431-445.
- Cohomology of locally nilpotent groups, J. Pure Appl. Algebra 48(1987), 281-300.
- Homology and cohomology of locally supersoluble groups, Math. Proc. Cambridge Philos. Soc. 102 (1987), 233-250.
- A finiteness condition on automorphism groups, (with F. Menogazzo), Rend. Sem. Mat. Univ. Padova 78 (1987), 267-277.
- Vanishing theorems for cohomology of locally nilpotent groups, Group Theory, pp. 120-129, Lecture Notes in Math., vol. 1281, Springer, Berlin (1987).
- Groups whose quotients have finite derived subgroup, (with Z. Zhang), J. Algebra 118 (1988), 346-368.
- Some remarks on central automorphisms of hypercentral groups, (with M. Curzio, H. Smith and J. Wiegold), Arch. Math. (Basel), 53 (1989), 327-331.
- Cohomology in infinite group theory, Proceedings of 1987 Singapore Group Theory Conference, pp. 29-53, de Gruyter, Berlin (1989).
- Recent progress on rewritability in groups, (with R.D. Blyth), in Proceedings of 1987 Singapore Group Theory Conference, pp. 77-85, de Gruyter, Berlin (1989).
- Reflections on the constructive theory of polycyclic groups, Groups-Korea 1988, Springer Lecture Notes, vol. 1398 (1989), 163-169.
- Solution of the solvability problem for rewritable groups, (with R.D. Blyth), J. London Math. Soc. (2) 41 (1990), 438-444.
- Algorithmic problems for automorphisms and endomorphisms of infinite soluble groups, Algebraic Structures and Number Theory, pp. 247-254, World Scientific, Singapore 1990.
- Deciding if an automorphism of an infinite soluble group is inner, Glasgow Math. J. 32 (1990), 265-272.
- A note on local coboundaries for locally nilpotent groups, Proceedings of conference on combinatorial group theory, College Park, 1988, Contemp. Math. 109 (1990), 159-163.
- The algorithmic theory of polycyclic-by-finite groups, (with G. Baumslag, F.B. Cannonito and D. Segal), J. Algebra 142 (1991), 118-149.
- Insoluble groups with the rewriting property P_8, (with R.D. Blyth), J. Pure Appl. Algebra 72 (1991), 251-263.
- On the structure of the normal subgroups of a group: nilpotency, (with J.C. Beidleman), Forum Math. 3 (1991), 581-593.
- Algorithms for finitely generated soluble groups, Lecture Note Series, National University of Singapore 39 (1992).
- On the structure of the normal subgroups of a group: supersolubility, (with J.C. Beidleman), Rend. Sem. Mat. Univ. Padova 87 (1992), 139-149.
- Triple factorizations by abelian groups, (with S.E. Stonehewer), Arch. Math., 60 (1993), 223-232.
- Imperfect Groups, (with A.J. Berrick), J. Pure Appl. Algebra. 88 (1993), 3-22.
- Groups with triple factorizations by abelian groups, (with S.E. Stonehewer), Proceedings of Conference on Infinite Groups and Group Rings 1992, pp. 122-126, World Scientific, Singapore 1993.
- The algorithmic theory of finitely generated metabelian groups, (with G. Baumslag and F.B. Cannonito), Trans. Amer. Math. Soc. 344 (1994), 629-648.
- Semisimple algebras, Galois actions and group cohomology, (with E. Aljadeff), J. Pure Appl. Algebra 94 (1994), 1-15.
- On subsemigroups and submonoids of a group, (with W. Yang), Ricerche di Mat. 43 (1994), 231-237.
- Review of Products of Groups by B. Amberg, S. Franciosi and F. de Giovanni, Bull. Amer. Math. Soc. 30 (1994), 262-268.
- The generalized Wielandt subgroup of a group, (with J.C. Beidleman and M. Dixon), Canad. J. Math. 47 (1995), 246-261.
- Solution of the twisting problem for skew group algebras, (with E. Aljadeff), Israel J. Math. 91 (1995), 409-417.
- Automorphisms fixing every subnormal subgroup of a finite group, Arch. Math. (Basel) 64 (1995), 1-4.
- Semisimple groups with the rewriting property Q_8, (with R.D. Blyth), Commun. in Algebra 23 (1995), 2171- 2180.
- Normalizers of subnormal subgroups, Infinite Groups 94, Editors F. Giovanni and M. Newell, de Gruyter, Berlin 1995.
- A topological characterization of free abelian groups, (with M. Timm), Proc. Workshop Geometric Topology 1994, Brigham Young Univ. Press 1996.
- Virtually trivial automorphisms of finitely generated groups, (with A. M. Hassanabadi and J. Wiegold), Bull. Iran. Math. Soc. 22 (1996), 35-40.
- On finite groups satisfying the permutizer condition, (with J.C. Beidleman), J. Algebra 191 (1997), 686-703.
- Groups in which every automorphism centralizes a subgroup of prime index, (with A. M. Hassanabadi and J. Wiegold), Commun. in Algebra 25 (10), (1997), 3119-3136.
- An algebraic model for the representation of accounting systems, (with R.A. Nehmer), Ann. Operations Research 71 (1997), 179-198.
- On groups that are isomorphic with every subgroup of finite index and their topology, (with M. Timm), J. London Math. Soc. (2)57 (1998), 91-104.
- The permutizer condition in infinite soluble groups, (with J.C. Beidleman), J. Algebra 210 (1998), 311-319.
- Permutability properties of subgroups, Groups St. Andrews 1997 in Bath., London Math. Soc. Lecture Notes vol. 261 (1999), 633-638.
- A survey of groups in which normality or permutability is a transitive relation, Algebra: Recent Advances, pp. 171-181, New Delhi (1999).
- Criteria for permutability to be transitive in finite groups, (with J.C. Beidleman and B. Brewster), J. of Algebra 222 (1999), 400-412.
- The Wielandt subgroup, Encyclopedia of Math., Supplement II, Kluwer, Dordrecht, pp. 509-510, 2000.
- On chief factors of dinilpotent groups, (with S.E. Stonehewer), J. Algebra, 234(2000), 480-491.
- The structure of finite groups in which permutability is transitive, J. Austral. Math. Soc. 70 (2001), 143-159.
- Derivations and the permutability of subgroups in polycyclic-by-finite groups, Proc. Amer. Math. Soc. 130 (2002), 3461-3464.
- Minimality and groups in which Sylow permutability is transitive, Ukrain. Mat. Z. 54 (2002), 856-865 = Ukr. Math. Soc. 54(2002), 1038-1049.
- Finite groups in which normality or permutability is transitive, Advances in Group Theory. Proceedings of Conference in group theory, University of Naples, 2002. Aracne. Rome 2003.
- Finite groups whose cyclic subnormal subgroups are permutable, Algebra Colloquium 12 (2005), 171-180.
- The maximal and minimal conditions for normal subgroups of infinite order or index, (with F. de Giovanni, D.H. Paek and A. Russo), Commun. in Algebra. 33 (2005), 183-199.
- On finite minimal non-nilpotent groups. (with A. Ballester- Bolinches and R. Esteban-Romero), Proc. Amer. Math. Soc. 133 (2005), 3455-3469.
- On co-hopfian groups, (with G. Endimioni), Public. Math. Debrecen 67 (2005), 423-436.
- Groups with finitely many derived subgroups, (with F. de Giovanni), J. London Math. Soc. 71 (2005), 658-668.
- The Maier-Schmidt problem for infinite groups, (with M.R. Celentani and A. Leone), J. Algebra 301 (2006), 294-307.
- On inert subgroups of a group. Rend. Sem. Mat. Univ. Padova 115 (2006), 137-159.
- On groups which contain no HNN-extensions, (with A. Russo and G. Vincenzi), Internat. J. Algebra and Computation 17 (2007), 1377-1387.
- Five Lectures on the Theory of Infinite Groups, Lectures given at Chengdu University of Information Science and Technology, Urbana, IL 2007.
- On groups in which every subgroup is closed in the profinite topology, (with A. Russo and G. Vicenzi), J. Pure Appl. Algebra 213 (2009), 421-429.
- Generalized Baumslag-Solitar groups and geometric homomorphisms, (with A.L. Delgado and M. Timm), J. Pure Appl. Algebra 213 (2009), 421-429.
- Sylow permutability in locally finite groups, Ricerche di Mat. 59 (2010), 313-318
- Recent results on generalized Baumslag-Solitar groups, Advances in Group Theory, Note di Matematica 30 (2010), 1-17.
- On the theory of generalized FC-groups, (with A. Russo and G. Vincenzi), J. of Algebra,326 (2011), 218-226.
- The Schur multiplier of a generalized Baumslag-Solitar group, Rend. Sem. Mat. Univ. Padova, 125 (2011), 207-215.
- Permutability and seriality in locally finite groups, Ischia Group Theory 2010, pp. 284--291, World Scientific, Hackensack, New Jersey, 2012.
- On groups with two isomorphism classes of derived subgroups, (with P. Longobardi, M. Maj and H. Smith), Glasgow Math. J. 55 (2013), 655-668..
- Automorphisms of group extensions, Note di Matematica 33 (2013), 121-129.
- Locally finite groups with finitely many isomorphism classes of derived subgroups, (with P. Longobardi and M.Maj), J. Algebra 393 (2013), 102-119.
- Generalized Baumslag-Solitar graphs with soluble fundamental groups, (with A. Delgado and M. Timm), Algebra Colloquium 21 (2014), 53-58.
- Recent results on groups with few isomorphism classes of derived subgroups, (with P. Longobardi and M. Maj), Contemp. Math. 611 (2014), 121-135.
- Group algebras with the bounded splitting property, (with B. Torrecillas), Commun. in Algebra 42 (2014), 5381-5386.
- Systems of companies with assets in common, (with S. Cruz Rambaud), Physica A 416 (2015), 129-134.
- Generalized Baumslag-Solitar groups: a survey of recent progress, Groups St. Andrews 2013, London Math. Soc Lecture Notes 423 (2015), pp. 457-468.
- On the cohomology of finite soluble groups, Arch. Math. (Basel) 105 (2015), 101-108.
- On the structure of subsets of an orderable group with some small doubling properties, (with G.A. Freiman, M. Herzog,, P. Longobardi, A. Plagne and Y.V. Stanchescu), J. Algebra 445 (2016), 307-326.
- On groups with extreme centralizers and normalizers, Advances in Group Theory and Applications. 1 (2016), 125-140.
- Cyclic normal subgroups of generalized Baumslag-Solitar groups, (with A.L. Delgado and M. Timm), Commun. in Algebra 45 (2017), 1808-1818.
- Sylow permutability in generalized soluble groups, J. Group Theory 20 (2017), 61-70.
- Sylow permutability in soluble minimax groups, Ricerche di Mat. 66 (2017), 521-530.
- 3-manifolds and generalized Baumslag-Solitar groups, (with A.L. Delgado and M. Timm), Commun. in Analysis and Geometry, 26 (2018), 571-584.
- On groups with countably many maximal subgroups, (with A. Arikan and G. Cutolo), J. Group Theory 21 (2018), 253-271.
- On groups that are dominated by countably many proper subgroups, (with A. Arikan and G. Cutolo), J. Algebra 509 (2018), 445-466.
- On metanilpotent groups satisfying the minimal condition on normal subgroups, J. Group Theory 22 (2019), 809--836.
- On infinite groups whose finite quotients have restricted prime divisors, Internat. J. Group Theory, 10(2021), 75-88.
- The seriality problem for weakly Sylow permutable subgroups in locally finite groups. J. Algebra 567(2021), 229-242.
- Finitely generated metabelian groups arising from integer polynomials, J. Group Theory, 27(2024), 313-343.