# Tour:Multiplicative group modulo n

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This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)PREVIOUS: Multiplicative monoid modulo n|UP: Introduction four (beginners)|NEXT: Elements of multiplicative group equal generators of additive group

General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part

WHAT YOU NEED TO DO:

- Understand the definition stated below.
- Convince yourself of the explanation that
invertibilityis the same as beingrelatively primeto .- Go through the examples, and convince yourself of facts (1) and (2).

## Definition

Let be a positive integer. The **multiplicative group modulo ** is the subgroup of the multiplicative monoid modulo n comprising the elements that have inverses.

Equivalently, it is the group, under multiplication, of elements in that are relatively prime to . (The two definitions are equivalent because if and are relatively prime, there exist integers such that , so ).

## Facts

- The order of the multiplicative group modulo equals the number of elements in that are relatively prime to . This number is termed the Euler-phi function or Euler totient function of , and is denoted .
- For a prime , . In other words, every nonzero element less than is invertible modulo .
- The multiplicative group modulo is a cyclic group if and only if for an odd prime and a natural number.

This page is part of the Groupprops guided tour for beginners. Make notes of any doubts, confusions or comments you have about this page before proceeding.PREVIOUS: Multiplicative monoid modulo n|UP: Introduction four (beginners)|NEXT: Elements of multiplicative group equal generators of additive group