Let o(G) be the average order of elements G, where G is a finite group. We show that there no polynomial lower bound for in terms o(N), N⊴G, even when prime-power group and N abelian. This gives negative answer to question A. Jaikin-Zapirain.

This paper concerns the enumeration of simultaneous conjugacy classes [Formula: see text]-tuples commuting matrices in upper triangular group text] and unitriangular over finite field odd characteristic. is done for text], by computing branching rules. Further, using matrix thus computed, we explicitly get probabilities each case.