Let $ G $ be a group. A subset $ S $ of $ G $ is said to normally generate $ G $ if the normal closure of $ S $ in $ G $ is equal to $ G $ itself. This means that every element of $ G $ can be represented as a product of conjugates of elements of $ S $ and their inverses. Given an element $ g $ of $ G $ and a normally generating set $ S, $ we define the length of $ g $ with respect to $ S $ as the smallest number of conjugates of elements of $ S $ or their inverses needed to express $ g $ as a product. Then, for each such $ S $, the diameter of $ G $ with respect to $ S $ is defined as the supremum of the lengths of elements of $ G $ with respect to $ S. $ The conjugacy diameter of $ G $ is the supremum of all diameters of $ G $ over all finite normally generating subsets. It measures how efficiently $ G $ is normally generated by its finite normally generating subsets.
In this paper, we found the conjugacy diameters of finite dihedral groups. It is worth noting that the conjugacy diameters of other families, such as semidihedral $ 2 $-groups, generalized quaternion groups, and modular $ p $-groups, have already been investigated.
Citation: Fawaz Aseeri. The conjugation diameters of finite dihedral groups[J]. AIMS Mathematics, 2025, 10(11): 27277-27289. doi: 10.3934/math.20251199
Let $ G $ be a group. A subset $ S $ of $ G $ is said to normally generate $ G $ if the normal closure of $ S $ in $ G $ is equal to $ G $ itself. This means that every element of $ G $ can be represented as a product of conjugates of elements of $ S $ and their inverses. Given an element $ g $ of $ G $ and a normally generating set $ S, $ we define the length of $ g $ with respect to $ S $ as the smallest number of conjugates of elements of $ S $ or their inverses needed to express $ g $ as a product. Then, for each such $ S $, the diameter of $ G $ with respect to $ S $ is defined as the supremum of the lengths of elements of $ G $ with respect to $ S. $ The conjugacy diameter of $ G $ is the supremum of all diameters of $ G $ over all finite normally generating subsets. It measures how efficiently $ G $ is normally generated by its finite normally generating subsets.
In this paper, we found the conjugacy diameters of finite dihedral groups. It is worth noting that the conjugacy diameters of other families, such as semidihedral $ 2 $-groups, generalized quaternion groups, and modular $ p $-groups, have already been investigated.
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Fawaz Aseeri. The conjugation diameters of finite dihedral groups[J]. AIMS Mathematics, 2025, 10(11): 27277-27289. doi: 10.3934/math.20251199
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