The problem of finding the number of ordered commuting tuples of elements in a finite group is equivalent to finding the size of the solution set of the system of equations determined by the commutator relations that impose commutativity among any pair of elements from an ordered tuple. We consider this type of systems for the case of ordered triples and express the size of the solution set in terms of the irreducible characters of the group. The obtained formulas are natural extensions of Frobenius’ character formula that calculates the number of ways a group element is a commutator of an ordered pair of elements in a finite group. We discuss how our formulas can be used to study the probability distributions afforded by these systems of equations, and we show explicit calculations for dihedral groups.

中文翻译：

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有限群中的换向器方程

求有限群中元素的有序交换元组数量的问题等价于求由交换子关系确定的方程组的解集的大小，该交换子关系在有序元组中的任何一对元素之间施加交换性。我们在有序三元组的情况下考虑这种类型的系统，并根据群的不可约特征来表示解集的大小。得到的公式是 Frobenius 字符公式的自然扩展，该公式计算了一个群元素是有限群中有序元素对的交换子的方式数。我们讨论了如何使用我们的公式来研究这些方程组提供的概率分布，并展示了二面体群的显式计算。