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A281854
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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).
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4
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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, 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
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OFFSET
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1,1
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COMMENTS
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The length of row n is given in A281855.
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.
See also the W. Lang links for these groups.
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LINKS
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EXAMPLE
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The triangle T(n, k) begins (N = A033949(n)):
n, N, phi(N)\ k 1 2 3 4 ...
1, 8, 4: 2 2
2, 12, 4: 2 2
3, 15, 8: 4 2
4, 16, 8: 4 2
5, 20, 8: 4 2
6, 21, 12: 3 2 2
7, 24, 8: 2 2 2
8, 28, 12: 3 2 2
9, 30, 8: 4 2
10, 32, 16: 8 2
11, 33, 20: 5 2 2
12, 35, 24: 4 3 2
13, 36, 12: 3 2 2
14, 39, 24: 4 3 2
15, 40, 16: 4 2 2
16, 42, 12: 3 2 2
17, 44, 20: 5 2 2
18, 45, 24: 4 3 2
19, 48, 16: 4 2 2
20, 51, 32: 16 2
21, 52, 24: 4 3 2
22, 55, 40: 5 4 2
23, 56, 24: 3 2 2 2
24, 57, 36: 9 2 2
25, 60, 16: 4 2 2
...
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>.
(In the Wikipedia Table <2, 20> is used).
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n = 32, A033949(32) = N = 70, phi(70) = 24.
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).
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.
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.
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.
The second 6-cycle from the powers of, say, 31 contains also the 6-cycle from 61.
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)
The cycle graph of C_4 X C_3 x C_2 is the 7th entry of Figure 4 of this link.
(End)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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