SPECIAL MAXIMAL SUBGROUPS OF \$p\$-GROUPS

In the 2006 edition of the Kourovka Notebook, Berkovich poses the following problem (Problem 16.13): Let \(p\) be a prime and \(P\) be a finite \(p\)-group. Can \(P\) have every maximal subgroup special? We show that the structure of such groups is very restricted, but for all primes there are groups of arbitrarily large size with this property. DOI: 10.1017/S0004972713000658