On strongly flat p-groups

In this paper, we provide a survey (both theoretical and computational) of the conditions for determining which finite p -groups are strongly flat. Let m(G) denote the maximum size of an irredundant generating set of a group G . Then, a group G is strongly flat if m(G) is strictly greater than m(H) for all proper subgroups H of G . For example, J. Whiston has shown in his dissertation that all symmetric groups are strongly flat. Here, we shall only consider the simpler case of finite p -groups. We observe that non-abelian powerful p -groups exhibit this property only when p is odd. Meanwhile, computations made using GAP show that, in general, the occurrences of strongly flat groups vary greatly between the cases where p is odd and where p = 2 . When p is odd, there are very few strongly flat p -groups as the group must be nilpotent of class 2 and have exponent p or p2 . When p = 2 , strongly flat p -groups are fairly common, as both the class and exponent of the group are unbounded. In particular, over 1 4 of the groups of order 256 are found to be strongly flat. These concepts are of interest in a number of situations: Powerful p -groups are important in the study of profinite groups, and also, many computations of m(G) are aided by knowing that some maximal subgroups are strongly flat. Throughout the paper, we shall mainly be following the notations and definitions of [6].