# Tour:Elements of multiplicative group equal generators of additive group

**This article adapts material from the main article:** elements of multiplicative group equal generators of additive group

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)PREVIOUS: Multiplicative group modulo n|UP: Introduction four (beginners)|NEXT: Multiplicative group modulo a prime is cyclic

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WHAT YOU NEED TO DO:

- Carefully understand the statement below, and try proving it yourself.
- Need the proof and make sure you understand it.

## Statement

Let be a positive integer, and consider the group of integers modulo n. Then, an element in this is a *generator* for the group of integers modulo if and only if it is an element of the multiplicative group modulo .

Note that these are also the same as the elements in that are relatively prime to , and the number of such elements is .

## Proof

### Generator of additive group implies element of multiplicative group

If is a generator of , then some integer multiple of must be equal to the element . Thus, there exists such that in . Viewing as a congruence class modulo , we see that is invertible modulo , and hence is in the multiplicative group.

### Element of multiplicative group implies generator of additive group

If is a element of the multiplicative group modulo , there exists an integer such that . Thus, the cyclic subgroup containing must also contain . But any subgroup containing must equal the whole group , so generates the whole group.

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)PREVIOUS: Multiplicative group modulo n|UP: Introduction four (beginners)|NEXT: Multiplicative group modulo a prime is cyclic

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