It has been proven that if a finite group is generated by a subset S, then each group element may be expressed as a word from the alphabet S of length less than or equal to the order of the group.Every finite group is finitely generated since 〈G〉 = G.While it is true that every quotient of a finitely generated group is finitely generated (simply take the images of the generators in the quotient), a subgroup of a finitely generated group need not be finitely generated.For example, let G be the free group in two generators, x and y (which is clearly finitely generated, since G = 〈〉), and let S be the subset consisting of all elements of G of the form y, for n a natural number.In this case, 〈x〉 is the cyclic subgroup of the powers of x, a cyclic group, and we say this group is generated by x.Equivalent to saying an element x generates a group is saying that 〈x〉 equals the entire group G.
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