FINITENESS CRITERIA FOR COVERINGS OF GROUPS BY FINITELY MANY SUBGROUPS OR COSETS

Yıl 2007, Cilt: 2 Sayı: 2, 83 - 89, 01.12.2007

Mira Bhargava

Öz

We prove that, for any positive integer n, there exists a minimal finite set S(n) of finite groups such that: a group G is the union of n of its proper subgroups (but not the union of fewer than n proper subgroups) if and only if G has a quotient isomorphic to some group K ∈ S(n). We prove, furthermore, that such a minimal finite set S(n) is in fact unique up to isomorphism of its members. Finally, an analogue of this result can be proved when “subgroups” is replaced more generally by “cosets”.

Anahtar Kelimeler

finite groups, unions of subgroups, unions of cosets, coverings of groups, minimal covers

Kaynakça

• M. Bhargava, Problem E1592, Amer. Math. Monthly, 71 (1964), 319.
• M. Bhargava, When is a group the union of proper normal subgroups? Amer. Math. Monthly, 109(5) (2002), 471–473.
• M. Bhargava, Groups as unions of subgroups, Amer. Math. Monthly, to appear. M. A. Brodie, R. F. Chamberlain, and L.-C. Kappe, Finite coverings by normal subgroups, Proc. Amer. Math. Soc., 104(3) (1988), 179–188.
• M. A. Berger, A. Felzenbaum, and A. Fraenkel, The Herzog-Schnheim conjec- ture for Şnite nilpotent groups, Canad. Math. Bull., 29(3) (1986), 329–333.
• R. A. Bryce, V. Fedri and L. Serena, Covering groups with subgroups, Bull. Austral. Math. Soc., 55(3) (1997), 469–476.
• R. A. Bryce, V. Fedri and L. Serena, A generalized Hughes property of Şnite groups, Comm. Algebra, 31(9) (2003), 4215–4243.
• J. H. E. Cohn, On n-sum groups, Math. Scand., 75 (1994), 44–58.
• P. Erdos, On integers of the form 2k+ p and some related problems, Summa Brasil. Math., 2 (1950), 113–123.
• P. E. Holmes, Subgroup coverings of some sporadic groups, preprint. M. S. Lucido, On the covers of Şnite groups, Groups St. Andrews, vol. II (2001), 395–399.
• A. Mar´oti, Covering the symmetric groups with proper subgroups, Journal of Combinatorial Theory Ser. A, 110 (1) (2005), 97–111.
• B. H. Neumann, Groups covered by Şnitely many cosets, Publ. Math. Debre- cen, 3 (1954), 227–242 (1955).
• M. M. Parmenter, Exact covering systems for groups, Fund. Math. 123(2) (1984), 133–136.
• M. M. Parmenter, Finite coverings by cosets of normal subgroups, Proc. Amer. Math. Soc., 110 (4) (1990), 877–880.
• Z. Sun, On the Herzog-Schonheim conjecture for uniform covers of groups, J. Algebra, 273 (1) (2004), 153–175.
• M. J. Tomkinson, Groups as the union of proper subgroups, Math. Scand., (2) (1997), 191–198.
• M. J. Tomkinson, Groups covered by Şnitely many cosets or subgroups, Comm. Algebra, 15(4) (1987), 845–859.
• G. Zappa, The papers of Gaetano Scorza on group theory (Italian), Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. (9) Mat. Appl., 2(2) (1991), 95–101. Mira Bhargava
• Department of Mathematics Hofstra University Hempstead, NY 11550
• E-mail: [e-mail]