A group generated by a single element g consists of the (not necessarily distinct) positive and negative powers of g:

... g^-2, g^-1, g⁰=e, g¹, g², ...

The group (Z,+,0) or (Z/nZ,+,0) is cyclic. It can be generated by 1 and by -1.

The group (Z/10Z)^* of invertible elements in Z/10Z is cyclic. It can be generated by the element 3.

Every element of S_n is a product of transpositions, see Section 5.2. Thus S_n is generated by its transpositions.

The even elements of S_n can be written as products of 3-cycles, see Section 5.3. Hence A_n is generated by its 3-cycles.

Consider D_n the group of symmetries of a regular n-gon. If r and s denote two reflections in D_n whose reflection lines make an angle of /n, then their product is a rotation over 2/n. Hence we have the following equalities where e denotes the identity map.

r² = e = s²,

(rs)ⁿ = e

Now it is straightforward to check that the elements of D_n are

e, r, rs, rsr, ..., rsr ··· sr = s.

(Can you find out which one of these is a reflection and which one is a rotation?) So, the group D_n is generated by r and s.