Counting Subgroups in a Direct Product of Finite Cyclic Groups

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The College Mathematics Journal. ) earned his Ph.D.from Binghamton University in 2003. He is currently an associate professor of mathematics and holds the endowed Cole Chair in Mathematics at Alfred University in beautiful western New York. His main research focuses on subgroup properties in finite groups. He enjoys teaching, problem solving, playing guitar, and enjoying life with his wife, Lynn, and his son, Julian.

Subgroups of Finite Abelian Groups Having Rank Two via Goursat’s Lemma

“…The results of Theorems 4.1 and 4.3 generalize and put in more compact forms those of G. Cȃlugȃreanu [4], J. Petrillo [13] and M. Tȃrnȃuceanu [15], obtained for p-groups of rank two, and included in Corollaries 4.2 and 4.4. We remark that both the papers [4] and [13] applied Goursat's lemma for groups (the first one in a slightly different form), while the paper [15] used a different approach based on properties of certain attached matrices.…”

“…In the paper [7] the identities (5), (6), (10), (13) and (14) were derived using another approach. The identity (13), as a special case of a formula valid for arbitrary finite abelian groups, was obtained by the author [16,17] using different arguments.…”

“…The identity (13), as a special case of a formula valid for arbitrary finite abelian groups, was obtained by the author [16,17] using different arguments. Finally, we remark that the functions (m, n) → s(m, n) and (m, n) → c(m, n) are multiplicative, viewed as arithmetic functions of two variables.…”

“…We remark that both the papers [4] and [13] applied Goursat's lemma for groups (the first one in a slightly different form), while the paper [15] used a different approach based on properties of certain attached matrices.…”

“…For the history, proof, discussion, applications and a generalization of Goursat's lemma see [1,2,5,9,12,13]. Corollary 2.2 is given in [5,Cor.…”