Diameter of a direct power of a finite group

ABSTRACT

We present two conjectures concerning the diameter of a direct power of a finite group. The first conjecture states that the diameter of Gⁿ with respect to each generating set is at most n(|G|−rank(G)); and the second one states that there exists a generating set 𝒜, of minimum size, for Gⁿ such that the diameter of Gⁿ with respect to 𝒜 is at most n(|G|−rank(G)). We will establish evidence for each of the above mentioned conjectures.

KEYWORDS:

• Diameter
• direct power
• finite group
• generating set
• rank

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20B05
• 20B30
• 20D60

Acknowledgments

The author wishes to express her thanks to her supervisors for suggesting the problem and for many stimulating conversations.

Notes

¹Usually A⊆G is considered to be a generating set, if every element of G can be expressed as a sequence of elements in A∪A⁻¹. When G is finite the definitions coincide.