�ber den Frobenius'schen Klassenbegriff in nilpotenten Gruppen

ON FINITE p-GROUPS WITH SUBGROUPS OF BREADTH 1

“…Moreover, G 2 is a Dedekind group; if it were hamiltonian then again (G ) 2 ≤ (Z (G 2 )) 2 = 1, hence the commutator equalities just found would yield the contradiction (G 2 ) = 1. Therefore G 2 is abelian; the same equalities now show that…”

“…Suppose that exp Z = exp G, let z be an element of Z of maximal order pq and let S = z q . Then |(G/S) | = p 2 and there exists g ∈ G Z such that z q / ∈ [g, G], because otherwise br(G/S) = 1 and hence |(G/S) | = p by a result in [2]. …”

“…The primary aim of this note is [2] On finite p-groups with subgroups of breadth 1 85 to record the fact that the derived subgroup of such a group is extremely constrained in size. Indeed, our main result is the following.…”

The structure of finite groups with three class sizes

“…Finally the structure stated in the theorem will be completely established when we prove that jP 0 j ¼ p and jP=ZðGÞ p j ¼ p 2 . The first claim follows easily from the fact that the class sizes of P are f1; pg (see [14], for instance). On the other hand, P 0 is an abelian normal subgroup of P of index p, so we have P ¼ P 0 h yi ¼ P 0 C G ð yÞ for any y A P À P 0 .…”

Commutator subgroups of finite p-groups

“…The number b(P) is called the breadth of P (see [3], [8]) and has important connections with the nilpotency class of P ( [3], [4] …”