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The classification of finite groups by using iteration digraphs. (English) Zbl 1413.05132

Summary: A digraph is associated with a finite group by utilizing the power map \({f:G\rightarrow G}\) defined by \(f(x)=x^{k}\) for all \(x\in G\), where \(k\) is a fixed natural number. It is denoted by \(\gamma_{G}(n,k)\). In this paper, the generalized quaternion and 2-groups are studied. The height structure is discussed for the generalized quaternion. The necessary and sufficient conditions on a power digraph of a 2-group are determined for a 2-group to be a generalized quaternion group. Further, the classification of two generated 2-groups as abelian or non-abelian in terms of semi-regularity of the power digraphs is completed.

MSC:

05C20 Directed graphs (digraphs), tournaments
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
20D15 Finite nilpotent groups, \(p\)-groups
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References:

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