FINITE p-GROUPS ALL OF WHOSE MAXIMAL SUBGROUPS, EXCEPT ONE, HAVE ITS DERIVED SUBGROUP OF ORDER ≤ p

Let G be a finite p-group which has exactly one maximal subgroup H such that |H| > p. Then we have d(G) = 2, p = 2, H is a four-group, G is abelian of order 8 and type (4, 2), G is of class 3 and the structure of G is completely determined. This solves the problem Nr. 1800 stated by Y. Berkovich in [3]. We consider here only finite p-groups and our notation is standard (see [1]). If G is a p-group all of whose maximal subgroups have its derived subgroups of order ≤ p, then such groups G are characterized in [3, §137]. But there is no way to determine completely the structure of such p-groups. It is quite surprising that we can determine completely (in terms of generators and relations) the title groups, where exactly one maximal subgroup has the commutator subgroup of order > p. We shall prove our main theorem (Theorem 8) starting with some partial results about the title groups. However, Propositions 4 and 6 are also of independent interest. Proposition 1. Let G be a title p-group. Then we have d(G) ≤ 3, cl(G) ≤ 3, p ≤ |G| ≤ p and G is abelian of exponent ≤ p. Also, G has at most one abelian maximal subgroup. Proof. Let H be the unique maximal subgroup of G with |H | > p. This gives |G| ≥ p. LetK 6= L be maximal subgroups ofG which are both distinct from H . We have |K | ≤ p, |L| ≤ p and so K L ≤ Z(G) and |K L| ≤ p. By a result of A. Mann ([1, Exercise 1.69]), we get |G : (K L)| ≤ p. This implies that |G| ≤ p, G is abelian and G is of class ≤ 3. Since K L is elementary 2010 Mathematics Subject Classification. 20D15.