OFFSET
1,7
COMMENTS
The multiplicative group of integers modulo n, (Z/n*Z)^x, also the cyclotomic group, the Galois group Gal(Q(zeta(n))/Q) with zeta(n) = exp(2*Pi*I/n), is cyclic for n from A033948 and non-cyclic for n from A033949. Each of these groups is the direct product of cyclic factors (one factor is included).
In the total factorization for n >= 3 only cyclic factors whose orders are prime powers appear, and because the direct product is associative, and for these abelian groups also commutative, one can order the factors with nonincreasing orders.
For n=1 and n=2 the group is C_1 = {1} (for n=1 one has 1 == 0 (mod 1)).
Cyclic groups may also have a factorization into more than one factor. E.g., C_6 = C_3 x C_2.
The number of factors in this total factorization is for a cyclic group C_m, for m >= 2, given by A001221(m). For m=1 this number is 1 (not A001221(1)).
For non-cyclic groups the number of factors in this total factorization is given by A281855(m) if n = A033949(m), m >= 1.
For the non-cyclic group case see also the W. Lang links under A281854.
EXAMPLE
CROSSREFS
KEYWORD
nonn
AUTHOR
Wolfdieter Lang, Mar 02 2017
STATUS
approved