Multiplicative group of integers modulo n

In modular arithmetic, the integers coprime (relatively prime) to n from the set of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n. Hence another name is the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n. Here units refers to elements with a multiplicative inverse, which in this ring are exactly those coprime to n.

This group, usually denoted , is fundamental in number theory. It has found applications in cryptography, integer factorization, and primality testing. It is an abelian, finite group whose order is given by Euler's totient function: For prime n the group is cyclic and in general the structure is easy to describe, though even for prime n no general formula for finding generators is known.

Group axioms

It is a straightforward exercise to show that, under multiplication, the set of congruence classes modulo n that are coprime to n satisfy the axioms for an abelian group.

Indeed, a is coprime to n if and only if gcd(a, n) = 1. Integers in the same congruence class ab (mod n) satisfy gcd(a, n) = gcd(b, n), hence one is coprime to n if and only if the other is. Thus the notion of congruence classes modulo n that are coprime to n is well-defined.

Since gcd(a, n) = 1 and gcd(b, n) = 1 implies gcd(ab, n) = 1, the set of classes coprime to n is closed under multiplication.

Integer multiplication respects the congruence classes, that is, aa' and bb' (mod n) implies aba'b' (mod n). This implies that the multiplication is associative, commutative, and that the class of 1 is the unique multiplicative identity.

Finally, given a, the multiplicative inverse of a modulo n is an integer x satisfying ax ≡ 1 (mod n). It exists precisely when a is coprime to n, because in that case gcd(a, n) = 1 and by Bézout's lemma there are integers x and y satisfying ax + ny = 1. Notice that the equation ax + ny = 1 implies that x is coprime to n, so the multiplicative inverse belongs to the group.

Notation

The set of (congruence classes of) integers modulo n with the operations of addition and multiplication is a ring. It is denoted   or    (the notation refers to taking the quotient of integers modulo the ideal or consisting of the multiples of n). Outside of number theory the simpler notation is often used, though it can be confused with the p-adic integers when n is a prime number.

The multiplicative group of integers modulo n, which is the group of units in this ring, may be written as (depending on the author)         (for German Einheit, which translates as unit), , or similar notations. This article uses

The notation refers to the cyclic group of order n. It is isomorphic to the group of integers modulo n under addition. Note that or may also refer to the group under addition. For example, the multiplicative group for a prime p is cyclic and hence isomorphic to the additive group , but the isomorphism is not obvious.

Structure

The order of the multiplicative group of integers modulo n is the number of integers in coprime to n. It is given by Euler's totient function: (sequence A000010 in the OEIS). For prime p, .

Cyclic case

The group is cyclic if and only if n is 1, 2, 4, pk or 2pk, where p is an odd prime and k > 0. For all other values of n the group is not cyclic.[1][2][3] This was first proved by Gauss.[4]

This means that for these n:

where

By definition, the group is cyclic if and only if it has a generator g, that is, the powers give all possible residues modulo n coprime to n (the first powers give each exactly once). A generator of is called a primitive root modulo n.[5] If there is any generator, then there are of them.

Powers of 2

Modulo 1 any two integers are congruent, i.e., there is only one congruence class, [0], coprime to 1. Therefore, is the trivial group with φ(1) = 1 element. Because of its trivial nature, the case of congruences modulo 1 is generally ignored and some authors choose not to include the case of n = 1 in theorem statements.

Modulo 2 there is only one coprime congruence class, [1], so is the trivial group.

Modulo 4 there are two coprime congruence classes, [1] and [3], so the cyclic group with two elements.

Modulo 8 there are four coprime congruence classes, [1], [3], [5] and [7]. The square of each of these is 1, so the Klein four-group.

Modulo 16 there are eight coprime congruence classes [1], [3], [5], [7], [9], [11], [13] and [15]. is the 2-torsion subgroup (i.e., the square of each element is 1), so is not cyclic. The powers of 3, are a subgroup of order 4, as are the powers of 5,   Thus

The pattern shown by 8 and 16 holds[6] for higher powers 2k, k > 2: is the 2-torsion subgroup (so is not cyclic) and the powers of 3 are a cyclic subgroup of order 2k 2, so

General composite numbers

By the fundamental theorem of finite abelian groups, the group is isomorphic to a direct product of cyclic groups of prime power orders.

More specifically, the Chinese remainder theorem[7] says that if then the ring is the direct product of the rings corresponding to each of its prime power factors:

Similarly, the group of units is the direct product of the groups corresponding to each of the prime power factors:

For each odd prime power the corresponding factor is the cyclic group of order , which may further factor into cyclic groups of prime-power orders. For powers of 2 the factor is not cyclic unless k = 0, 1, 2, but factors into cyclic groups as described above.

The order of the group is the product of the orders of the cyclic groups in the direct product. The exponent of the group, that is, the least common multiple of the orders in the cyclic groups, is given by the Carmichael function (sequence A002322 in the OEIS). In other words, is the smallest number such that for each a coprime to n, holds. It divides and is equal to it if and only if the group is cyclic.

Subgroup of false witnesses

If n is composite, there exists a subgroup of the multiplicative group, called the "group of false witnesses", in which the elements, when raised to the power n − 1, are congruent to 1 modulo n (since the residue 1, to any power, is congruent to 1 modulo n, the set of such elements is nonempty).[8] One could say, because of Fermat's Little Theorem, that such residues are "false positives" or "false witnesses" for the primality of n. The number 2 is the residue most often used in this basic primality check, hence 341 = 11 × 31 is famous since 2340 is congruent to 1 modulo 341, and 341 is the smallest such composite number (with respect to 2). For 341, the false witnesses subgroup contains 100 residues and so is of index 3 inside the 300 element multiplicative group mod 341.

Examples

n = 9

The smallest example with a nontrivial subgroup of false witnesses is 9 = 3 × 3. There are 6 residues coprime to 9: 1, 2, 4, 5, 7, 8. Since 8 is congruent to −1 modulo 9, it follows that 88 is congruent to 1 modulo 9. So 1 and 8 are false positives for the "primality" of 9 (since 9 is not actually prime). These are in fact the only ones, so the subgroup {1,8} is the subgroup of false witnesses. The same argument shows that n − 1 is a "false witness" for any odd composite n.

n = 91

For n = 91 (= 7 x 13), there are residues coprime to 91, half of them (i.e., 36 of them) are false witnesses of 91, namely 1, 3, 4, 9, 10, 12, 16, 17, 22, 23, 25, 27, 29, 30, 36, 38, 40, 43, 48, 51, 53, 55, 61, 62, 64, 66, 68, 69, 74, 75, 79, 81, 82, 87, 88, and 90, since for these values of x, x90 is congruent to 1 mod 91.

n = 561

n = 561 (= 3 x 11 x 17) is a Carmichael number, thus s560 is congruent to 1 modulo 561 for any integer s coprime to 561. The subgroup of false witnesses is, in this case, not proper; it is the entire group of multiplicative units modulo 561, which consists of 320 residues.

Examples

This table shows the cyclic decomposition of and a generating set for n ≤ 128. The generating sets are not unique; e.g., modulo 16 both {15, 3} and {15, 5} will work, in the list, we list the smallest values, (thus, for n with primitive root, we list the smallest primitive root modulo n) for example, for modulo 12, we list {5, 7} instead of {5, 11} or {7, 11} .

For example, we take n = 20. means that the order of is 8 (i.e., there are 8 numbers less than 20 and coprime to it); that is, the fourth power of any number coprime to 20 is congruent to 1 (mod 20); and as for the generators, 19 has (element) order 2, 3 has (element) order 4, and every member of is of the form 19a × 3b, where a is 0 or 1 and b is 0, 1, 2, or 3.

The powers of 19 are {±1} and the powers of 3 are {3, 9, 7, 1}. The latter and their negatives modulo 20, {17, 11, 13, 19} are all the numbers less than 20 and coprime to it. That the order of 19 is 2 and the order of 3 is 4 implies that the fourth power of every member of is congruent to 1 (mod 20).

Smallest primitive root mod n are (0 if no root exists)

0, 1, 2, 3, 2, 5, 3, 0, 2, 3, 2, 0, 2, 3, 0, 0, 3, 5, 2, 0, 0, 7, 5, 0, 2, 7, 2, 0, 2, 0, 3, 0, 0, 3, 0, 0, 2, 3, 0, 0, 6, 0, 3, 0, 0, 5, 5, 0, 3, 3, 0, 0, 2, 5, 0, 0, 0, 3, 2, 0, 2, 3, 0, 0, 0, 0, 2, 0, 0, 0, 7, 0, 5, 5, 0, 0, 0, 0, 3, 0, 2, 7, 2, 0, 0, 3, 0, 0, 3, 0, ... (sequence A046145 in the OEIS)

Numbers of the elements in a minimal generating set of mod n are

0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 3, 1, 2, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 3, 1, 1, 1, 3, 2, 1, 2, 3, 1, 2, ... (sequence A046072 in the OEIS)
Group structure of (Z/nZ)×
Generating set   Generating set   Generating set   Generating set
1 C1110 33 C2×C1020102, 10 65 C4×C1248122, 12 97 C9696965
2 C1111 34 C1616163 66 C2×C1020105, 7 98 C4242423
3 C2222 35 C2×C1224122, 6 67 C6666662 99 C2×C3060302, 5
4 C2223 36 C2×C61265, 19 68 C2×C1632163, 67 100 C2×C2040203, 99
5 C4442 37 C3636362 69 C2×C2244222, 68 101 C1001001002
6 C2225 38 C1818183 70 C2×C1224123, 69 102 C2×C1632165, 101
7 C6663 39 C2×C1224122, 38 71 C7070707 103 C1021021025
8 C2×C2423, 5 40 C2×C2×C41643, 11, 39 72 C2×C2×C62465, 17, 19 104 C2×C2×C1248123, 5, 103
9 C6662 41 C4040406 73 C7272725 105 C2×C2×C1248122, 29, 41
10 C4443 42 C2×C61265, 13 74 C3636365 106 C5252523
11 C1010102 43 C4242423 75 C2×C2040202, 74 107 C1061061062
12 C2×C2425, 7 44 C2×C1020103, 43 76 C2×C1836183, 37 108 C2×C1836185, 107
13 C1212122 45 C2×C1224122, 44 77 C2×C3060302, 76 109 C1081081086
14 C6663 46 C2222225 78 C2×C1224125, 7 110 C2×C2040203, 109
15 C2×C4842, 14 47 C4646465 79 C7878783 111 C2×C3672362, 110
16 C2×C4843, 15 48 C2×C2×C41645, 7, 47 80 C2×C4×C43243, 7, 79 112 C2×C2×C1248123, 5, 111
17 C1616163 49 C4242423 81 C5454542 113 C1121121123
18 C6665 50 C2020203 82 C4040407 114 C2×C1836185, 37
19 C1818182 51 C2×C1632165, 50 83 C8282822 115 C2×C4488442, 114
20 C2×C4843, 19 52 C2×C1224127, 51 84 C2×C2×C62465, 11, 13 116 C2×C2856283, 115
21 C2×C61262, 20 53 C5252522 85 C4×C1664162, 3 117 C6×C1272122, 17
22 C1010107 54 C1818185 86 C4242423 118 C58585811
23 C2222225 55 C2×C2040202, 21 87 C2×C2856282, 86 119 C2×C4896483, 118
24 C2×C2×C2825, 7, 13 56 C2×C2×C62463, 13, 29 88 C2×C2×C1040103, 5, 7 120 C2×C2×C2×C43247, 11, 19, 29
25 C2020202 57 C2×C1836182, 20 89 C8888883 121 C1101101102
26 C1212127 58 C2828283 90 C2×C1224127, 11 122 C6060607
27 C1818182 59 C5858582 91 C6×C1272122, 3 123 C2×C4080407, 83
28 C2×C61263, 13 60 C2×C2×C41647, 11, 19 92 C2×C2244223, 91 124 C2×C3060303, 61
29 C2828282 61 C6060602 93 C2×C30603011, 61 125 C1001001002
30 C2×C4847, 11 62 C3030303 94 C4646465 126 C6×C63665, 13
31 C3030303 63 C6×C63662, 5 95 C2×C3672362, 94 127 C1261261263
32 C2×C81683, 31 64 C2×C1632163, 63 96 C2×C2×C83285, 17, 31 128 C2×C3264323, 127

See also

Notes

  1. Weisstein, Eric W. "Modulo Multiplication Group". MathWorld.
  2. Primitive root, Encyclopedia of Mathematics
  3. (Vinogradov 2003, pp. 105–121, § VI PRIMITIVE ROOTS AND INDICES)
  4. (Gauss & Clarke 1986, arts. 52–56, 82–891)
  5. (Vinogradov 2003, p. 106)
  6. (Gauss & Clarke 1986, arts. 90–91)
  7. Riesel covers all of this. (Riesel 1994, pp. 267–275)
  8. Erdős, Paul; Pomerance, Carl (1986). "On the number of false witnesses for a composite number". Math. Comput. 46 (173): 259–279. doi:10.1090/s0025-5718-1986-0815848-x. Zbl 0586.10003.

References

The Disquisitiones Arithmeticae has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.

  • Gauss, Carl Friedrich; Clarke, Arthur A. (translator into English) (1986), Disquisitiones Arithmeticae (Second, corrected edition), New York: Springer, ISBN 978-0-387-96254-2
  • Gauss, Carl Friedrich; Maser, H. (translator into German) (1965), Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition), New York: Chelsea, ISBN 978-0-8284-0191-3
  • Riesel, Hans (1994), Prime Numbers and Computer Methods for Factorization (second edition), Boston: Birkhäuser, ISBN 978-0-8176-3743-9
  • Vinogradov, I. M. (2003), "§ VI PRIMITIVE ROOTS AND INDICES", Elements of Number Theory, Mineola, NY: Dover Publications, pp. 105–121, ISBN 978-0-486-49530-9
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