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Abstract
We introduce the normalizing graph of a group and study the connectivity of the normalizing and permuting graphs of a group when the group is finite and soluble. In particular, we classify finite soluble groups with disconnected normalizing graph. The main results shows that if a finite soluble group has connected normalizing graph then this graph has diameter at most 6. Furthermore, this bound is tight. A corollary then presents the connectivity properties of the permuting graph.
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