  A finite group $G$ is called *uniformly generated*, if whenever there is a
(strictly ascending) chain of subgroups $1<\langle x_1\rangle<\langle
x_1,x_2\rangle <\cdots<\langle x_1,x_2,\dots,x_d\rangle=G$, then $d$ is the
minimal number of generators of $G$. Our main result classifies the uniformly
generated groups without using the simple group classification. These groups
are related to finite projective geometries by a result of Iwasawa on subgroup
lattices.
