OFFSET
2,1
COMMENTS
Every finite abelian group can be uniquely expressed as the direct product: C_n1 X C_n2 X ... X C_ns for some integers n1,n2,...,ns where each integer is greater than 1 and each successive integer divides its predecessor. The integers n1,n2,...,ns are called the invariant factors. The order of the group is the product of its invariant factors.
REFERENCES
D. S. Dummit and R. M. Foote, Abstract Algebra, Wiley, 2003, 3rd Edition, page 158.
LINKS
Alois P. Heinz, Rows n = 2..5000, flattened
EXAMPLE
{2},
{3},
{4}, {2, 2},
{5},
{6},
{7},
{8}, {4, 2}, {2, 2, 2},
{9}, {3, 3},
{10},
{11},
{12}, {6, 2},
{13},
{14},
{15},
{16}, {8, 2}, {4, 4}, {4, 2, 2}, {2, 2, 2, 2}
{17},
{18}, {6, 3},
{19},
{20}, {10, 2},
{21},
{22},
{23},
{24}, {12, 2}, {6, 2, 2},
{25}, {5, 5},
{26},
{27}, {9, 3}, {3, 3, 3},
{28}, {14, 2},
{29},
{30},
The row corresponding to n = 12 is 12,6,2 because the invariant factor decompositions of the 2, A000688(12), abelian groups of order 12 are: C_12 and C_6 X C_2
MATHEMATICA
f[{x_, y_}] := x^IntegerPartitions[y];
g[n_] := FactorInteger[n][[1, 1]];
h[list_] :=Apply[Times, Map[PadRight[#, Max[Map[Length, SplitBy[list, g]]], 1] &, SplitBy[list, g]]];
Table[Map[h, Join @@@ Tuples[Map[f, FactorInteger[n]]]], {n, 2,
30}] // Grid
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Geoffrey Critzer, Nov 25 2015
STATUS
approved