Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The number of isotype and $ l$-pure subgroups of an abelian $ p$-group


Author: Paul Hill
Journal: Proc. Amer. Math. Soc. 32 (1972), 69-74
MSC: Primary 20K99
DOI: https://doi.org/10.1090/S0002-9939-1972-0291282-0
MathSciNet review: 0291282
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The number of isotype subgroups of an abelian p-group G is determined. This solves a recent problem of Fuchs. Actually, we accomplish slightly more. Define a subgroup H of an abelian p-group G to be an l-pure subgroup of G if, for some ordinal $ \lambda $, H is $ {p^\lambda }$-pure in G and $ {p^\lambda }H$ is divisible. We compute the number of l-pure subgroups of G and show that the number of l-pure subgroups and the number of isotype subgroups of G coincide. Our final result deals with the number of nonisomorphic isotype subgroups of G when G is a direct sum of countable groups.


References [Enhancements On Off] (What's this?)

  • [1] D. Boyer, A note on a problem of Fuchs, Pacific J. Math. 10 (1960), 1147. MR 22 #9533. MR 0118762 (22:9533)
  • [2] L. Fuchs, Abelian groups, Publishing House of the Hungarian Academy of Sciences, Budapest, 1958. MR 21 #5672. MR 0106942 (21:5672)
  • [3] -, Infinite abelian groups. Vol. 1, Pure and Appl. Math., vol. 36, Academic Press, New York, 1970. MR 41 #333.
  • [4] P. Hill, On the number of pure subgroups, Pacific J. Math. 12 (1962), 203-205. MR 25 #3999. MR 0140581 (25:3999)
  • [5] -, Sums of countable primary groups, Proc. Amer. Math. Soc. 17 (1966), 1469-1470. MR 33 #7408. MR 0199259 (33:7408)
  • [6] -, Isotype subgroups of direct sums of countable groups, Illinois J. Math. 13 (1969), 281-290. MR 39 #1550. MR 0240198 (39:1550)
  • [7] P. Hill and C. Megibben, On primary groups with countable basic subgroups, Trans. Amer. Math. Soc. 124 (1966), 49-59. MR 33 #7409. MR 0199260 (33:7409)
  • [8] J. Irwin, C. Walker and E. Walker, On $ {p^\alpha }$-pure sequences of Abelian groups, Topics in Abelian Groups (Proc. Sympos., New Mexico State Univ., 1962), Scott, Foresman, Chicago, Ill., 1963, pp. 69-119. MR 33 #7410. MR 0199261 (33:7410)
  • [9] G. Kolettis, Jr., Direct sums of countable groups, Duke Math. J. 27 (1960), 111-125. MR 22 #1616. MR 0110748 (22:1616)
  • [10] R. Nunke, Purity and subfunctors of the identity, Topics in Abelian Groups (Proc. Sympos., New Mexico State Univ., 1962), Scott, Foresman, Chicago, Ill., 1963, pp. 121-171. MR 30 #156. MR 0169913 (30:156)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 20K99

Retrieve articles in all journals with MSC: 20K99


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0291282-0
Keywords: Abelian p-group, pure subgroups, isotype subgroups, direct sum of countable groups
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society