Copulas are real functions representing the dependence structure of the distribution of a random vector, and measures of concordance associate with every copula a numerical value in order to allow for the comparison of different degrees of dependence. We first introduce and study a group of transformations mapping the collection of all copulas of fixed but arbitrary dimension into itself. These transformations may be used to construct new copulas from a given one or to prove that certain real functions on the unit cube are indeed copulas. It turns out that certain transformations of a symmetric copula may be asymmetric, and vice versa. Applying this group, we then propose a concise definition of a measure of concordance for copulas. This definition, in which the properties of a measure of concordance are defined in terms of two particular subgroups of the group, provides an easy access to the investigation of invariance properties of a measure of concordance. In particular, it turns out that for copulas which are invariant under a certain subgroup the value of every measure of concordance is equal to zero. We also show that the collections of all transformations which preserve symmetry or the concordance order or the value of every measure of concordance each form a subgroup and that these three subgroups are identical. Finally, we discuss a class of measures of concordance in which every element is defined as the expectation with respect to the probability measure induced by a fixed copula having an invariance property with respect to two subgroups of the group. This class is rich and includes the well-known examples Spearman's rho and Gini's gamma.

Copulas are real functions representing the dependence structure of the distribution of a random vector, and measures of concordance associate with every copula a numerical value in order to allow for the comparison of different degrees of dependence. We first introduce and study a group of transformations mapping the collection of all copulas of fixed but arbitrary dimension into itself. These transformations may be used to construct new copulas from a given one or to prove that certain real functions on the unit cube are indeed copulas. It turns out that certain transformations of a symmetric copula may be asymmetric, and vice versa. Applying this group, we then propose a concise definition of a measure of concordance for copulas. This definition, in which the properties of a measure of concordance are defined in terms of two particular subgroups of the group, provides an easy access to the investigation of invariance properties of a measure of concordance. In particular, it turns out that for copulas which are invariant under a certain subgroup the value of every measure of concordance is equal to zero. We also show that the collections of all transformations which preserve symmetry or the concordance order or the value of every measure of concordance each form a subgroup and that these three subgroups are identical. Finally, we discuss a class of measures of concordance in which every element is defined as the expectation with respect to the probability measure induced by a fixed copula having an invariance property with respect to two subgroups of the group. This class is rich and includes the well-known examples Spearman's rho and Gini's gamma.