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A281854 Irregular triangle read by rows. Row n gives the orders of the cyclic groups appearing as factors in the direct product decomposition of the abelian non-cyclic multiplicative groups of integers modulo A033949(n).

%I A281854 #38 Nov 13 2017 02:58:57
%S A281854 2,2,2,2,4,2,4,2,4,2,3,2,2,2,2,2,3,2,2,4,2,8,2,5,2,2,4,3,2,3,2,2,4,3,
%T A281854 2,4,2,2,3,2,2,5,2,2,4,3,2,4,2,2,16,2,4,3,2,5,4,2,3,2,2,2,9,2,2,4,2,2
%N A281854 Irregular triangle read by rows. Row n gives the orders of the cyclic groups appearing as factors in the direct product decomposition of the abelian non-cyclic multiplicative groups of integers modulo A033949(n).
%C A281854 The length of row n is given in A281855.
%C A281854 The multiplicative group of integers modulo n is written as (Z/(n Z))^x (in ring notation, group of units) isomorphic to Gal(Q(zeta(n))/Q) with zeta(n) = exp(2*Pi*I/n). The present table gives in row n the factors of the direct product decomposition of the non-cyclic group of integers modulo A033949(n) (in nonincreasing order). The cyclic group of order n is C_n. Note that only C-factors of prime power orders are used; for example C_6 has the decomposition C_3 x C_2, etc. C_n is decomposed whenever n has relatively prime factors like in C_30 = C_15 x C_2 = C_5 x C_3 x C_2. In the Wikipedia table partial decompositions appear.
%C A281854 The row products phi(A033949(n)) are given as 4*A281856(n), n >= 1, with phi(n) = A000010(n).
%C A281854 See also the W. Lang links for these groups.
%H A281854 Wolfdieter Lang, <a href="https://arxiv.org/abs/1210.1018">The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon</a>, Table 7 (in row n = 80 it should read Z_4^2 x Z_2), arXiv:1210.1018 [math.GR], 2012.
%H A281854 Wolfdieter Lang, <a href="/A282624/a282624.pdf">Table for the multiplicative non-cyclic groups of integers modulo A033949</a>.
%H A281854 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n ">Multiplicative group of integers modulo n </a>. Compare with the Table at the end.
%e A281854 The triangle T(n, k) begins (N = A033949(n)):
%e A281854 n,   N, phi(N)\ k  1  2  3  4 ...
%e A281854 1,   8,   4:       2  2
%e A281854 2,  12,   4:       2  2
%e A281854 3,  15,   8:       4  2
%e A281854 4,  16,   8:       4  2
%e A281854 5,  20,   8:       4  2
%e A281854 6,  21,  12:       3  2  2
%e A281854 7,  24,   8:       2  2  2
%e A281854 8,  28,  12:       3  2  2
%e A281854 9,  30,   8:       4  2
%e A281854 10, 32,  16:       8  2
%e A281854 11, 33,  20:       5  2  2
%e A281854 12, 35,  24:       4  3  2
%e A281854 13, 36,  12:       3  2  2
%e A281854 14, 39,  24:       4  3  2
%e A281854 15, 40,  16:       4  2  2
%e A281854 16, 42,  12:       3  2  2
%e A281854 17, 44,  20:       5  2  2
%e A281854 18, 45,  24:       4  3  2
%e A281854 19, 48,  16:       4  2  2
%e A281854 20, 51,  32:      16  2
%e A281854 21, 52,  24:       4  3  2
%e A281854 22, 55,  40:       5  4  2
%e A281854 23, 56,  24:       3  2  2  2
%e A281854 24, 57,  36:       9  2  2
%e A281854 25, 60,  16:       4  2  2
%e A281854 ...
%e A281854 n = 6, A033949(6) = N = 21, phi(21) = 12, group (Z/21 n)^x decomposition C_3 x C_2 x C_2 (in the Wikipedia Table C_2 x C_6). The smallest positive reduced system modulo 21 has the primes {2, 5, 11, 13, 17, 19} with cycle lengths {6, 6, 6, 2, 6, 6}, respectively. As generators of the group one can take <2, 13>.
%e A281854   (In the Wikipedia Table <2, 20> is used).
%e A281854 ----------------------------------------------
%e A281854 From _Wolfdieter Lang_, Feb 04 2017: (Start)
%e A281854 n = 32, A033949(32) = N = 70, phi(70) = 24.
%e A281854 Cycle types (multiplicity as subscript): 12_7, 6_4, 4_2, 3_1, 2_2 (a total of 16 cycles). Cycle structure: 12_2, 6_2 (all other cycles are sub-cycles).
%e A281854 The first 12-cycle obtained from the powers of, say 3, contains also the 12-cycles from 17 and 47. It also contains the 4-cycle from 13, the 3-cycle from 11 and the 2-cycle from 29.
%e A281854 The second 12-cycle from the powers of, say, 23 contains also the 12-cycles from 37, 53 and 67, as well as the 4-cycle from 43.
%e A281854 The first 6-cycle from the powers of, say, 19 contains also the 6-cycle of 59 as well as the 2-cycle from 41.
%e A281854 The second 6-cycle from the powers of, say, 31 contains also the 6-cycle from 61.
%e A281854 The group is C_6 x C_4 = (C_2 x C_3) x C_4 = C_4 X C_3 x C_2 (see the W. Lang link, Table 7)
%e A281854 The cycle graph of C_4 X C_3 x C_2 is the 7th entry of Figure 4 of this link.
%e A281854 (End)
%Y A281854 Cf. A033949, A192005, A281855, A282624.
%K A281854 nonn,tabf
%O A281854 1,1
%A A281854 _Wolfdieter Lang_, Feb 02 2017