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# Smallest numbers of element

Let be the smallest number of elements necessary to generate a finite group . Show that .

(by convention if .)

Solution

We are using induction on . Clearly if then . The result is also true if since the non-identity element has order at least 2. Hence . Let . Assume that if a group   is generated by elements, then . Let the generators of be . Then the subgroup

is a proper subgroup of and by assumption . Since we obtain that is a left coset of in and . Moreover , . Hence