In this paper groups are considered inducing groups of power automorphisms on each factor of their derived series. In particular, it is proved that soluble groups with such property have derived length at most 3, and that this bound is best possible.
In [T. W. Müller, An arithmetic theorem related to groups of bounded nilpotency class, J. Algebra 300 (2006), 10-15] T. W. Müller characterizes the positive integers n satisfying the property that every group of order n is nilpotent of class bounded by a fixed positive integer c. In this article a different proof of the above result will be given.
Nel 1916 Issai Schur provò che se si colora l'insieme con un numero finito di colori, allora esistono dei numeri , e aventi lo stesso colore tali che . Egli utilizzò tale risultato nello studio della cosiddetta ``versione locale'' dell'Ultimo Teorema di Fermat dimostrando che se è un numero intero positivo, allora esiste un primo ``sufficientemente grande'' tale che l'equazione congruenziale ha una soluzione intera non banale. In quest'articolo si fornirà un'esposizione elementare...
The structure of infinite groups in which any two (proper) subgroups of the same cardinality are isomorphic is described within the universe of locally graded groups. The corresponding problem for finite groups was considered by R. Armstrong (1958).
This paper studies groups G whose all subgroups are either ascendant or self-normalizing. We characterize the structure of such G in case they are locally finite. If G is a hyperabelian group and has the property, we show that every subgroup of G is in fact ascendant provided G is locally nilpotent or non-periodic. We also restrict our study replacing ascendant subgroups by permutable subgroups, which of course are ascendant [Stonehewer S.E., Permutable subgroups of infinite groups, Math. Z., 1972,...
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