A264809 Irregular array read by rows: row n contains the invariant factors of every (up to isomorphism) abelian group of order n for n>=2.

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

Offset: 2

Author

Geoffrey Critzer, Nov 25 2015

Keywords

• nonn
• tabf

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.

Examples

			{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
		

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

Crossrefs

Cf. A249770.

Programs

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