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WTThe units modulo n form a finite abelian group

Theorem·T61

The units modulo n form a finite abelian group under multiplication.

For with , is a group, with the residue-class multiplication restricted to the units.
In words
The units modulo n under multiplication form a finite abelian group.
Rests onF02F03F04F05F06F08F09F11F12F13F14A01A02A06A08
Never needed: F10 · A03 · A04 · A05 · A07 · A09 (computed from the citation graph, not asserted).

Proof

  1. 1
    Write . Throughout, multiplication of classes is associative and commutative with unity (T59).
  2. 2
    is an operation on (closure). Let with inverses : and (D117). Using commutativity and associativity (T59), , so has the inverse and lies in (D117). Hence restricts to a binary operation on .
  3. 3
    Associativity (G1). Inherited from the associativity of on (T59).
  4. 4
    Identity (G2). , since (D117); and for all (T59, unity).
  5. 5
    Inverses (G3). Given with (D117), commutativity gives , so is itself a unit (inverse ) and , with . So D038 holds; commutativity of (T59) makes abelian.
  6. 6
    Finite. (D117), which is finite since (T60); so is finite (L24).
∎

Remarks

This is the multiplicative group of . It is not a subgroup of any group already at hand, since is only a commutative monoid (non-units and lack inverses); the group axioms are checked directly. Its order is Euler's totient . That it is a finite group is exactly the hypothesis of L36, which then gives for every unit , that is Euler's theorem. For prime the group has order (L80). It is in fact cyclic (a primitive root exists), though that is not proved here.
Wikipedia ↗ZMod.instGroupUnits ↗

Used by

L79L80T62
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