Multiplicative group of integers modulo n

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Not to be confused with Integers modulo n.

In modular arithmetic the set of congruence classes relatively prime to the modulus number, say n, form a group under multiplication called the multiplicative group of integers modulo n. It is also called 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. (Units refers to elements with a multiplicative inverse.)

This group is fundamental in number theory. It has found applications in cryptography, integer factorization, and primality testing. For example, by finding the order of this group, one can determine whether n is prime: n is prime if and only if the order is n − 1.

Group axioms[edit]

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

Because ab (mod n) implies that gcd(a, n) = gcd(b, n), the notion of congruence classes modulo n which are relatively prime to n is well-defined.

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

The natural mapping from the integers to the congruence classes modulo n that takes an integer to its congruence class modulo n respects products. This implies that the class containing 1 is the unique multiplicative identity, and also the associative and commutative laws hold. In fact it is a ring homomorphism.

Given a, gcd(a, n) = 1, finding x satisfying ax ≡ 1 (mod n) is the same as solving ax + ny = 1, which can be done by Bézout's lemma. The x found will have the property that gcd(x, n) = 1.

Notation[edit]

The (quotient) ring of integers modulo n is denoted \mathbb{Z}/n\mathbb{Z}  or  \mathbb{Z}/(n)  (i.e., the ring of integers modulo the ideal n\mathbb{Z} = (n) consisting of the multiples of n) or by \mathbb{Z}_n (though the latter can be confused with the p-adic integers when n is a prime number). Depending on the author, its group of units may be written (\mathbb{Z}/n\mathbb{Z})^*,   (\mathbb{Z}/n\mathbb{Z})^\times,   \mathrm{U}(\mathbb{Z}/n\mathbb{Z}),   \mathrm{E}(\mathbb{Z}/n\mathbb{Z})   (for German Einheit, which translates as unit) or similar notations. This article uses (\mathbb{Z}/n\mathbb{Z})^\times.

The notation \mathrm{C}_n refers to the cyclic group of order n.

Structure[edit]

n = 1[edit]

Modulo 1 any two integers are congruent, i.e. there is only one congruence class. Every integer is relatively prime to 1. Therefore the single congruence class modulo 1 is relatively prime to the modulus, so (\mathbb{Z}/1\,\mathbb{Z})^\times \cong \mathrm{C}_1 is trivial. This implies that φ(1) = 1. Since the first power of any integer is congruent to 1 modulo 1, λ(1) is also 1.

Because of its trivial nature, the case of congruences modulo 1 is generally ignored. For example, the theorem "(\mathbb{Z}/n\mathbb{Z})^\times is cyclic if and only if φ(n) = λ(n)" implicitly includes the case n = 1, whereas the usual statement of Gauss's theorem "(\mathbb{Z}/n\mathbb{Z})^\times is cyclic if and only if n = 2, 4, any power of an odd prime or twice any power of an odd prime," explicitly excludes 1.

Powers of 2[edit]

Modulo 2 there is only one relatively prime congruence class, 1, so (\mathbb{Z}/2\mathbb{Z})^\times \cong \mathrm{C}_1 is the trivial group.

Modulo 4 there are two relatively prime congruence classes, 1 and 3, so (\mathbb{Z}/4\mathbb{Z})^\times \cong \mathrm{C}_2, the cyclic group with two elements.

Modulo 8 there are four relatively prime classes, 1, 3, 5 and 7. The square of each of these is 1, so (\mathbb{Z}/8\mathbb{Z})^\times \cong \mathrm{C}_2 \times \mathrm{C}_2, the Klein four-group.

Modulo 16 there are eight relatively prime classes 1, 3, 5, 7, 9, 11, 13 and 15. \{\pm 1, \pm 7\}\cong \mathrm{C}_2 \times \mathrm{C}_2, is the 2-torsion subgroup (i.e. the square of each element is 1), so (\mathbb{Z}/16\mathbb{Z})^\times is not cyclic. The powers of 3, \{1, 3, 9, 11\} are a subgroup of order 4, as are the powers of 5, \{1, 5, 9, 13\}.   Thus (\mathbb{Z}/16\mathbb{Z})^\times \cong \mathrm{C}_2 \times \mathrm{C}_4.

The pattern shown by 8 and 16 holds[1] for higher powers 2k, k > 2: \{\pm 1, 2^{k-1} \pm 1\}\cong \mathrm{C}_2 \times \mathrm{C}_2, is the 2-torsion subgroup (so (\mathbb{Z}/2^k\mathbb{Z})^\times is not cyclic) and the powers of 3 are a subgroup of order 2k − 2, so (\mathbb{Z}/2^k\mathbb{Z})^\times \cong \mathrm{C}_2 \times \mathrm{C}_{2^{k-2}}.

Powers of odd primes[edit]

For powers of odd primes pk the group is cyclic:[2]

 (\mathbb{Z}/p^k\mathbb{Z})^\times \cong \mathrm{C}_{p^{k-1}(p-1)} \cong \mathrm{C}_{\varphi(p^k)} .

General composite numbers[edit]

The Chinese remainder theorem[3] says that if \;\;n=p_1^{k_1}p_2^{k_2}p_3^{k_3}\dots, \; then the ring \mathbb{Z}/n\mathbb{Z} is the direct product of the rings corresponding to each of its prime power factors:

\mathbb{Z}/n\mathbb{Z} \cong \mathbb{Z}/{p_1^{k_1}}\mathbb{Z}\; \times \;\mathbb{Z}/{p_2^{k_2}}\mathbb{Z} \;\times\; \mathbb{Z}/{p_3^{k_3}}\mathbb{Z}\dots\;\;

Similarly, the group of units (\mathbb{Z}/n\mathbb{Z})^\times is the direct product of the groups corresponding to each of the prime power factors:

(\mathbb{Z}/n\mathbb{Z})^\times\cong (\mathbb{Z}/{p_1^{k_1}}\mathbb{Z})^\times \times (\mathbb{Z}/{p_2^{k_2}}\mathbb{Z})^\times  \times (\mathbb{Z}/{p_3^{k_3}}\mathbb{Z})^\times \dots\;.

Subgroup of false witnesses[edit]

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).[4] One could say, because of Fermat's Little Theorem, that such residues are "false positives" or "false witnesses" for the primality of n. 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[edit]
n = 9

The smallest example with a nontrivial subgroup of false witnesses is 9 = 3 × 3. There are 6 residues relatively prime 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 = 561

561 is a Carmichael number, thus n560 is congruent to 1 modulo 561 for any number n coprime to 561. Thus 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.

Properties[edit]

Order[edit]

The order of the group is given by Euler's totient function: | (\mathbb{Z}/n\mathbb{Z})^\times|=\varphi(n). This is the product of the orders of the cyclic groups in the direct product. (sequence A000010 in OEIS)

Exponent[edit]

The exponent is given by the Carmichael function \lambda(n), the least common multiple of the orders of the cyclic groups. Thus, \lambda(n) is the smallest number for a given n such that for each a relatively prime to n, a^{\lambda(n)} \equiv 1 \pmod n holds. (sequence A002322 in OEIS)

Generators[edit]

The group (\mathbb{Z}/n\mathbb{Z})^\times is cyclic if and only if its order \varphi(n) is equal to its exponent \lambda(n). This is the case when n is 2, 4, pk or 2pk, where p is an odd prime and k > 0. For all other values of n (except 1) the group is not cyclic.[5][6] The single generator in the cyclic case is called a primitive root modulo n.

Since all the (\mathbb{Z}/n\mathbb{Z})^\times, n ≤ 7 are cyclic, another way to state this is: If n < 8 then (\mathbb{Z}/n\mathbb{Z})^\times has a primitive root. If n ≥ 8 then (\mathbb{Z}/n\mathbb{Z})^\times has a primitive root unless n is divisible by 4 or by two distinct odd primes. All ns which have a primitive root are listed in OEISA033948.

In the general case there is one generator for each cyclic direct factor.

Examples[edit]

This table shows the cyclic decomposition of (\mathbb{Z}/n\mathbb{Z})^\times and a generating set for n ≤ 96. 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} . The generators are listed in the same order as the direct factors.

The numbers of the elements in generating set of n are

1, 1, 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, 2, 2, 2, 1, 2, 3, ... (sequence A046072 in OEIS)

For example, we take n = 20. \varphi(20)=8 means that the order of (\mathbb{Z}/20\mathbb{Z})^\times is 8 (i.e. there are 8 numbers less than 20 and coprime to it); \lambda(20)=4 that the fourth power of any number relatively prime to 20 is congruent to 1 (mod 20); and as for the generators, 19 has order 2, 3 has order 4, and every member of (\mathbb{Z}/20\mathbb{Z})^\times 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 (\mathbb{Z}/20\mathbb{Z})^\times is congruent to 1 (mod 20).

Smallest number of maximum order to mod n are

0, 1, 2, 3, 2, 5, 3, 3, 2, 3, 2, 5, 2, 3, 2, 3, 3, 5, 2, 3, 2, 7, 5, 5, 2, 7, 2, 3, 2, 7, 3, 3, 2, 3, 2, 5, 2, 3, 2, 3, 6, 5, 3, 3, 2, 5, 5, 5, 3, 3, 5, 7, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 7, 5, 5, 5, 2, 3, 2, 7, 3, 3, 2, 7, 2, 5, 3, 3, 2, 3, 3, 7, 2, 3, 11, 5, 2, 5, ... (sequence A111076 in OEIS)
Group Structure of (Z/nZ)×
n\; (\mathbb{Z}/n\mathbb{Z})^\times \varphi(n) \lambda(n)\; generating set   n\; (\mathbb{Z}/n\mathbb{Z})^\times \varphi(n) \lambda(n)\; generating set   n\; (\mathbb{Z}/n\mathbb{Z})^\times \varphi(n) \lambda(n)\; generating set
1 C1 1 1 0 33 C2×C10 20 10 2, 10 65 C4×C12 48 12 2, 12
2 C1 1 1 1 34 C16 16 16 3 66 C2×C10 20 10 5, 7
3 C2 2 2 2 35 C2×C12 24 12 2, 6 67 C66 66 66 2
4 C2 2 2 3 36 C2×C6 12 6 5, 19 68 C2×C16 32 16 3, 67
5 C4 4 4 2 37 C36 36 36 2 69 C2×C22 44 22 2, 68
6 C2 2 2 5 38 C18 18 18 3 70 C2×C12 24 12 3, 11
7 C6 6 6 3 39 C2×C12 24 12 2, 38 71 C70 70 70 7
8 C2×C2 4 2 3, 7 40 C2×C2×C4 16 4 3, 11, 39 72 C2×C2×C6 24 6 5, 7, 11
9 C6 6 6 2 41 C40 40 40 6 73 C72 72 72 5
10 C4 4 4 3 42 C2×C6 12 6 5, 13 74 C36 36 36 5
11 C10 10 10 2 43 C42 42 42 3 75 C2×C20 40 20 2, 74
12 C2×C2 4 2 5, 7 44 C2×C10 20 10 3, 43 76 C2×C18 36 18 3, 75
13 C12 12 12 2 45 C2×C12 24 12 2, 44 77 C2×C30 60 30 2, 76
14 C6 6 6 3 46 C22 22 22 5 78 C2×C12 24 12 5, 7
15 C2×C4 8 4 2, 14 47 C46 46 46 5 79 C78 78 78 3
16 C2×C4 8 4 3, 15 48 C2×C2×C4 16 4 5, 7, 47 80 C2×C4×C4 32 4 3, 7, 11
17 C16 16 16 3 49 C42 42 42 3 81 C54 54 54 2
18 C6 6 6 5 50 C20 20 20 3 82 C40 40 40 7
19 C18 18 18 2 51 C2×C16 32 16 5, 50 83 C82 82 82 2
20 C2×C4 8 4 3, 19 52 C2×C12 24 12 7, 51 84 C2×C2×C6 24 6 5, 11, 13
21 C2×C6 12 6 2, 20 53 C52 52 52 2 85 C4×C16 64 16 2, 3
22 C10 10 10 7 54 C18 18 18 5 86 C42 42 42 3
23 C22 22 22 5 55 C2×C20 40 20 2, 21 87 C2×C28 56 28 2, 86
24 C2×C2×C2 8 2 5, 7, 13 56 C2×C2×C6 24 6 3, 13, 29 88 C2×C2×C10 40 10 3, 5, 7
25 C20 20 20 2 57 C2×C18 36 18 2, 20 89 C88 88 88 3
26 C12 12 12 7 58 C28 28 28 3 90 C2×C12 24 12 7, 11
27 C18 18 18 2 59 C58 58 58 2 91 C6×C12 72 12 2, 3
28 C2×C6 12 6 3, 13 60 C2×C2×C4 16 4 7, 11, 19 92 C2×C22 44 22 3, 91
29 C28 28 28 2 61 C60 60 60 2 93 C2×C30 60 30 5, 11
30 C2×C4 8 4 7, 11 62 C30 30 30 3 94 C46 46 46 5
31 C30 30 30 3 63 C6×C6 36 6 2, 5 95 C2×C36 72 36 2, 3
32 C2×C8 16 8 3, 31 64 C2×C16 32 16 3, 63 96 C2×C2×C8 32 8 5, 7, 95

See also[edit]

Notes[edit]

  1. ^ Gauss, DA, arts. 90–91
  2. ^ Gauss, DA, arts. 52–56, 82–89
  3. ^ Riesel covers all of this. pp. 267–275
  4. ^ Erdős, Paul; Pomerance, Carl (1986). "On the number of false witnesses for a composite number". Math. Comput. 46: 259–279. doi:10.1090/s0025-5718-1986-0815848-x. Zbl 0586.10003. 
  5. ^ Weisstein, Eric W., "Modulo Multiplication Group", MathWorld.
  6. ^ Primitive root, Encyclopedia of Mathematics

References[edit]

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 0-387-96254-9 
  • 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 0-8284-0191-8 
  • Riesel, Hans (1994), Prime Numbers and Computer Methods for Factorization (second edition), Boston: Birkhäuser, ISBN 0-8176-3743-5 

External links[edit]