Book Review
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Book Information:
Author:
Roland Schmidt
Title:
Subgroup lattices of groups
Additional book information:
Expositions in Math., vol. 14, de Gruyter,
1994,
xv+572 pp.,
ISBN 3-11-011213-2,
$148.95$
R. W. Baddeley and A. Lucchini, On representing finite lattices as intervals in subgroup lattices of finite groups, preprint.
R. Baer, The significance of the system of subgroups for the structure of the group, Amer. J. Math. 61 (1939), 1–44.
J. J. Corliss, Upper limits to the real roots of a real algebraic equation, Amer. Math. Monthly 46 (1939), 334–338. MR 4
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
R. G. Burns and S. Oates-Williams, Varieties of groups and normal-subgroup lattices—a survey, Algebra Universalis 32 (1994), no. 1, 145–152. MR 1287020, DOI 10.1007/BF01190820
Richard Dedekind, Gesammelte mathematische Werke. Bände I–III, Chelsea Publishing Co., New York, 1968 (German). Herausgegeben von Robert Fricke, Emmy Noether und Öystein Ore. MR 0237282
—, Über die von drei Moduln erzeugte Dualgruppe, Math. Annalen 53 (1900), 371–403.
R. Freese, Finitely based modular congruence varieties are distributive, Algebra Universalis 32 (1994), no. 1, 104–114. MR 1287018, DOI 10.1007/BF01190818
Ralph Freese and Bjarni Jónsson, Congruence modularity implies the Arguesian identity, Algebra Universalis 6 (1976), no. 2, 225–228. MR 472644, DOI 10.1007/BF02485830
Marshall Hall Jr., The theory of groups, The Macmillan Company, New York, N.Y., 1959. MR 0103215
A. Yu. Ol’shanskiĭ, An infinite group with subgroups of prime orders, Inv. Akad. Nauk SSSR Ser. Mat. 44 (1980), 309–321 (Russian).
A. Yu. Ol′shanskiĭ, Groups of bounded period with subgroups of prime order, Algebra i Logika 21 (1982), no. 5, 553–618 (Russian). MR 721048
Irving Kaplansky, Infinite abelian groups, Revised edition, University of Michigan Press, Ann Arbor, Mich., 1969. MR 0233887
Peter Köhler, $M_{7}$ as an interval in a subgroup lattice, Algebra Universalis 17 (1983), no. 3, 263–266. MR 729935, DOI 10.1007/BF01194535
L. Lady, Finite Rank Torsion Free Modules over Dedekind Domains, preliminary version available at http://www.math.hawaii.edu/$\sim$lee.
Andrea Lucchini, Intervals in subgroup lattices of finite groups, Comm. Algebra 22 (1994), no. 2, 529–549. MR 1255880, DOI 10.1080/00927879408824862
Andrea Lucchini, Representation of certain lattices as intervals in subgroup lattices, J. Algebra 164 (1994), no. 1, 85–90. MR 1268327, DOI 10.1006/jabr.1994.1054
Ralph N. McKenzie, George F. McNulty, and Walter F. Taylor, Algebras, lattices, varieties. Vol. I, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1987. MR 883644
O. Ore, On the foundations of abstract algebra, I, Ann. Math. 36 (1935), 406–437.
—, On the foundations of abstract algebra, II, Ann. Math. 37 (1936), 265–292.
—, Structures and group theory, I, Duke Math. J. 3 (1937), 149–173.
—, Structures and group theory, II, Duke Math. J. 4 (1938), 247–269.
P. P. Pálfy, On Feit’s examples of intervals in subgroup lattices, J. Algebra 116 (1988), no. 2, 471–479. MR 953164, DOI 10.1016/0021-8693(88)90230-X
Péter Pál Pálfy and Pavel Pudlák, Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups, Algebra Universalis 11 (1980), no. 1, 22–27. MR 593011, DOI 10.1007/BF02483080
T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
P. P. Pálfy and Cs. Szabó, An identity for subgroup lattices of abelian groups, Algebra Universalis 33 (1995), no. 2, 191–195. MR 1318983, DOI 10.1007/BF01190930
E. Schröder, Algebra der Logik, Leipzig, 1890–1995, 3 volumes.
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
Jiří Tůma, Intervals in subgroup lattices of infinite groups, J. Algebra 125 (1989), no. 2, 367–399. MR 1018952, DOI 10.1016/0021-8693(89)90171-3
- R. W. Baddeley and A. Lucchini, On representing finite lattices as intervals in subgroup lattices of finite groups, preprint.
- R. Baer, The significance of the system of subgroups for the structure of the group, Amer. J. Math. 61 (1939), 1–44.
- —, A unified theory of projective spaces and finite Abelian groups, Trans. Amer. Math. Soc. 52 (1942), 283–343.
- G. Boole, An Investigation into the Laws of Thought, 1854, Reprinted by Open Court Publishing Co., Chicago, 1940. 01029
- R. G. Burns and S. Oates-Williams, Varieties of groups and normal-subgroup lattices—a survey, Algebra Universalis 32 (1994), 145–152.
- R. Dedekind, Über Zerlegungen von Zahlen durch ihre grössten gemeinsamen Teiler, Festschrift der Herzogl. technische Hochschule zur Naturforscher-Versammlung, Braunschweig (1897), Reprinted in “Gesammelte mathematische Werke”, Vol. 2, pp. 103–148, Chelsea, New York, 1968.
- —, Über die von drei Moduln erzeugte Dualgruppe, Math. Annalen 53 (1900), 371–403.
- R. Freese, Finitely based modular congruence varieties are distributive, Algebra Universalis 32 (1994), 104–114.
- R. Freese and B. Jónsson, Congruence modularity implies the Arguesian identity, Algebra Universalis 6 (1976), 225–228.
- M. Hall, The Theory of Groups, Chelsea, New York, 1959.
- A. Yu. Ol’shanskiĭ, An infinite group with subgroups of prime orders, Inv. Akad. Nauk SSSR Ser. Mat. 44 (1980), 309–321 (Russian).
- A. Yu. Ol’shanskiĭ, Groups of bounded period with subgroups of prime order, Algebra i Logica 21 (1982), 553–618 (Russian).
- I. Kaplansky, Infinite Abelian Groups, University of Michigan Press, Ann Arbor, 1969, Revised edition.
- P. Köhler, $M_7$ as an interval in a subgroup lattice, Algebra Universalis 17 (1983), 263–266.
- L. Lady, Finite Rank Torsion Free Modules over Dedekind Domains, preliminary version available at http://www.math.hawaii.edu/$\sim$lee.
- A. Lucchini, Intervals in subgroup lattices of finite groups, Communications in Algebra 22 (1994), 529–549.
- —, Representations of certain lattices as intervals in subgroup lattices, J. Algebra 164 (1994), 85–90.
- Ralph McKenzie, George McNulty, and Walter Taylor, Algebras, Lattices, Varieties, Volume I, Wadsworth and Brooks\slash Cole, Monterey, California, 1987.
- O. Ore, On the foundations of abstract algebra, I, Ann. Math. 36 (1935), 406–437.
- —, On the foundations of abstract algebra, II, Ann. Math. 37 (1936), 265–292.
- —, Structures and group theory, I, Duke Math. J. 3 (1937), 149–173.
- —, Structures and group theory, II, Duke Math. J. 4 (1938), 247–269.
- P. P. Pálfy, On Feit’s examples of intervals in subgroup lattices, J. Algebra 116 (1988), 471–479.
- P. P. Pálfy and P. Pudlák, Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups, Algebra Universalis 11 (1980), 22–27.
- P. P. Pálfy and C. Szabó, Congruence varieties of groups and Abelian groups, Lattice Theory and its Applications (K. A. Baker and R. Wille, eds.), Heldermann Verlag, Lemgo, Germany, 1995, In Celebration of Garrett Birkhoff’s 80th Birthday, Research and Exposition in Mathematics, Vol. 23, Darmstadt, Germany, June 13–17, 1991, pp. 163–184.
- —, An identity for subgroup lattices of Abelian groups, Algebra Universalis 33 (1995), 191–195.
- E. Schröder, Algebra der Logik, Leipzig, 1890–1995, 3 volumes.
- M. Suzuki, Structure of a Group and the Structure of its Lattice of Subgroups, Springer Verlag, Berlin, 1956.
- J. T\uuma, Intervals in subgroup lattices of infinite groups, J. Algebra 125 (1989), 367–399.
Review Information:
Reviewer:
Ralph Freese
Affiliation:
University of Hawaii
Email:
ralph@math.hawaii.edu
Journal:
Bull. Amer. Math. Soc.
33 (1996), 487-492
DOI:
https://doi.org/10.1090/S0273-0979-96-00676-3
Review copyright:
© Copyright 1996
American Mathematical Society
