The quotient of a generating set is a generating set of the quotient

Let G be a group and N \leq G a normal subgroup. Show that if G = \langle S \rangle, then G/N = \langle \overline{S} \rangle.


Let x \in G/N. Since G = \langle S \rangle, we have x = s_1^{a_1}s_2^{a_2} \ldots s_k^{a_k} for some s_i \in S and a_i \in \mathbb{Z}. Then xN = (s_1^{a_1}s_2^{a_2} \ldots s_k^{a_k})N = (s_1N)^{a_1}(s_2N)^{a_2} \ldots (s_kN)^{a_k} \in \langle \overline{S} \rangle.

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