×

Weak Cayley tables. (English) Zbl 0962.20003

If \(G\) is a finite group and \(\chi\) is a character of \(G\), then the \(2\)-character \(\chi^{(2)}\) is defined by \(\chi^{(2)}(g,h)=\chi(g)\chi(h)-\chi(gh)\). The aim of the paper is to find properties of \(G\) which can be determined by the \(1\)- and \(2\)-characters of the irreducible representations. The authors introduce the weak Cayley table to be the table whose rows and columns are indexed by the elements of \(G\) and the entry in the \((g,h)\)-th position is the conjugacy class of \(gh\). It is shown that \(1\)- and \(2\)-characters contain the same information as the weak Cayley table, and a class of \(2\)-groups which are determined by their weak Cayley tables is given. The main result and its generalizations examine group extensions, and are used to construct groups with the same weak Cayley table.

MSC:

20C15 Ordinary representations and characters
20D15 Finite nilpotent groups, \(p\)-groups
PDFBibTeX XMLCite
Full Text: DOI