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On absolutely-nilpotent of class k groups

Patrizia Longobardi, Trueman MacHenry, Mercede Maj, James Wiegold (1995)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

A group G in a variety V is said to be absolutely- V , and we write G A V , if central extensions by G are again in V . Absolutely-abelian groups have been classified by F. R. Beyl. In this paper we concentrate upon the class A N k of absolutely-nilpotent of class k groups. We prove some closure properties of the class A N k and we show that every nilpotent of class k group can be embedded in an A N k -gvoup. We describe all metacyclic A N k -groups and we characterize 2 -generator and infinite 3 -generator A N 2 -groups. Finally...

On connectedness of graphs on direct product of Weyl groups

Samy A. Youssef, S. G. Hulsurkar (1995)

Archivum Mathematicum

In this paper, we have studied the connectedness of the graphs on the direct product of the Weyl groups. We have shown that the number of the connected components of the graph on the direct product of the Weyl groups is equal to the product of the numbers of the connected components of the graphs on the factors of the direct product. In particular, we show that the graph on the direct product of the Weyl groups is connected iff the graph on each factor of the direct product is connected.

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