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## Main Question or Discussion Point

Hello,

I have a subgroup [itex]S=\left\langle A \right\rangle[/itex] generated by the set

When I need to prove by induction on

If the answer is n=0, then it seems to me that in most cases associated with "subgroups generated by some set", we always have to define n=0 as corresponding to the identity element, thus the basis of induction will be always about proving that some property holds for the identity element. Am I right?

I have a subgroup [itex]S=\left\langle A \right\rangle[/itex] generated by the set

*A*, i.e. [itex]S=\left\{ a_1 a_2 \ldots a_n \;|\; a_i \in A \right\}[/itex].When I need to prove by induction on

*n*some property of*S*, what should I choose as the base case of induction? n=1, or simply n=0 ?If the answer is n=0, then it seems to me that in most cases associated with "subgroups generated by some set", we always have to define n=0 as corresponding to the identity element, thus the basis of induction will be always about proving that some property holds for the identity element. Am I right?