Abstract. It is known that the Fejér means -- with respect to the Walsh and bounded Vilenkin systems -- of a continuous function converge everywhere to the function, and this convergence is uniform. That is, the celebrated result of Lipót Fejér is valid also for these systems. In this work we discuss problems concerning the above written on similar, not necessarily Abelian, totally disconnected groups with respect to the character system for functions that are constant on the conjugacy classes. We find that the nonabelian case completely differs from the commutative case. Even the theorem of Fejér fails to hold. We prove the existence of a $\gamma >0$, and $f\in{\rm Lip} (\gamma )$ Lipschitz function, such that $\sup |\sigma_nf|=+\infty $ on a dense set.

AMS Subject Classification (1991): 42C10

Keyword(s): noncommutative Vilenkin groups, character system, Fejér means, divergence, Lipschitz functions, Mathieu group

Received October 10, 2002, and in final form September 8, 2004. (Registered under 5865/2009.)