 Cauchy sample mean has an unchanged distribution

Independent standard Cauchy distribution variables sum to a Cauchy law of scale $n$, so their sample mean has its original distribution for every sample size. This follows by induction using convolution of probability densities. It does not contradict the usual weak law of large numbers, since the Cauchy law has no finite first absolute moment and its expectation is undefined. Symmetry does not turn a divergent expectation into zero.