Exercise 475Algebra › GroupsHigh schoolAdded 3 Nov 2026

Cyclic groups

#cyclic group#generator#examples

Statement

Show that each of the following groups is cyclic, and find a generator:

(a) $(\mathbb{Z}/6\mathbb{Z},+)$ ;
(b) $(\{1,i,-1,-i\},\times)$ ;
(c) $(\mathbb{Z}/5\mathbb{Z})^*,\times)$ .

Hint

A generator is an element whose powers (or multiples) give the whole group.

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(a) $\bar 1$ generates: $\bar 1,\bar 2,\bar 3,\bar 4,\bar 5,\bar 0$ . Also $\bar 5$ is a generator.

(b) $i$ generates: $i,i^2=-1,i^3=-i,i^4=1$ . Also $-i$ is a generator.

(c) $(\mathbb{Z}/5\mathbb{Z})^*=\{1,2,3,4\}$ under multiplication mod $5$ . $\bar 2$ generates: $2,4,3,1$ .

Conclusion. All three groups are cyclic.