Factorization numbers of finite abelian groups

Document Type : Research Paper


Ferdowsi University of Mashhad


The number of factorizations of a finite abelian group as the product of two subgroups is computed in two different ways and a combinatorial identity involving Gaussian binomial coefficients is presented‎.


Main Subjects

G. E. Andrews and K. Eriksson (2004). Integer Partitions. Cambridge University Press, Cambridge. W. N. Bailey (1941). A note on certain $q$-identities. Quart. J. Math.. 12, 173-175 J. N. S. Bidwell and M. J. Curran (2010). Automorphisms of finite Abelian groups. Math. Proc. R. Ir. Acad.. 110A (1), 57-71 H. Cohn (2004). Projective geometry over $\mathbb{F}_1$ and the Gaussian binomial coefficients. Amer. Math. Monthly. 111, 487-495 I. J. Davies (1962). Enumeration of certain subgroups of abelian $p$-groups. Proc. Edinb. Math. Soc. (2). 13, 1-4 S. Delsarte (1948). Fonctions de Mobius sur les groupes Abeliens finis. Ann. of Math. (2). 49, 600-609 M. Golasi\'{n}ski and D. L. Gon\c{c}alves (2008). On automorphisms of finite Abelian $p$-groups. Math. Slovaca. 58 (4), 405-412 C. J. Hillar and D. L. Rhea (2007). Automorphisms of finite abelian groups. Amer. Math. Monthly. 11, 17-23 N. J. S. Hughes (1951). The structure and order of the group of central automorphisms of a finite group. Proc. London Math. Soc.. 52 (2), 377-385 F. H. Jackson (1941). Certain $q$-identities. Quart. J. Math.. 12, 167-172 Y. Kinosita (1949). On an enumeration of certain subgroups of a $p$-group. J. Osaka Inst. Sci. Tech., Part I. 1, 13-20 J. Konvalina (1998). Generalized binomial coefficients and the subset-subspace problem. Adv. in Appl. Math.. 21, 228-240 M. W. Liebeck, C. E. Praeger and J. Saxl (1990). The maximal factorizations of the finite simple groups and their automorphism groups, iv+151 pp. Mem. Amer. Math. Soc.. 86 (432) G. A. Miller (1926). Form of the number of the prime power subgroups of an abelian group. Proc. Natl. Acad. Sci. USA. 12, 470-472 G. A. Miller (1939). Number of the subgroups of any given abelian group. Proc. Natl. Acad. Sci. USA. 25, 258-262 G. A. Miller (1939). Independent generators of the subgroups of an abelian group. Proc. Natl. Acad. Sci. USA. 25, 364-367 F. Saeedi and M. Farrokhi D. G. (2012). Factorization numbers of some finite groups. Glasgow J. Math.. 54, 345-354 F. Saeedi and M. Farrokhi D. G. Subgroup permutability degree of $PSL(2,p^n)$. to appear in Glasgow J. Math.. P. R. Sanders (1969). The central automorphisms of a finite group. J. London Math. Soc.. 44, 225-228 K. Shoda (1928). \"{U}ber die Automorphismen einer endlischen Abelschen Gruppe. Math. Ann.. 100, 674-686 T. Stehling (1992). On computing the number of subgroups of a finite Abelian group. Combinatorica. 12 (4), 475-479 M. T\v{a}rn\v{a}uceanu (2009). Subgroup commutativity degrees of finite groups. J. Algebra. 321 (9), 2508-2520 Y. Yeh (1948). On prime power abelian groups. Bull. Amer. Math. Soc.. 54, 323-327