单词 | Abelian Group | ||||||||||||||
释义 | Abelian GroupN.B. A detailed on-line essay by S. Finchwas the starting point for this entry. A Group for which the elements Commute (i.e., No general formula is known for giving the number of nonisomorphic Finite Groups of a givenOrder. However, the number of nonisomorphic Abelian Finite Groups
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![]() ![]() The Kronecker Decomposition Theorem states that every Finite Abelian group can be written as a DirectProduct of Cyclic Groups of Prime Power Orders. If the Orders of a Finite Group is a Prime Another interesting result is that if
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DeKoninck, J.-M. and Ivic, A. Topics in Arithmetical Functions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields. Amsterdam, Netherlands: North-Holland, 1980. Erdös, P. and Szekeres, G. ``Über die Anzahl abelscher Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem.'' Acta Sci. Math. (Szeged) 7, 95-102, 1935. Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/abel/abel.html Kendall, D. G. and Rankin, R. A. ``On the Number of Abelian Groups of a Given Order.'' Quart. J. Oxford 18, 197-208, 1947. Kolesnik, G. ``On the Number of Abelian Groups of a Given Order.'' J. Reine Angew. Math. 329, 164-175, 1981. Neumann, P. M. ``An Enumeration Theorem for Finite Groups.'' Quart. J. Math. Ser. 2 20, 395-401, 1969. Pyber, L. ``Enumerating Finite Groups of Given Order.'' Ann. Math. 137, 203-220, 1993. Richert, H.-E. ``Über die Anzahl abelscher Gruppen gegebener Ordnung I.'' Math. Zeitschr. 56, 21-32, 1952. Sloane, N. J. A. SequenceA000688/M0064in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Srinivasan, B. R. ``On the Number of Abelian Groups of a Given Order.'' Acta Arith. 23, 195-205, 1973. |
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