NEW REDUCTIONS AND LOGARITHMIC LOWER BOUNDS FOR THE NUMBER OF CONJUGACY CLASSES IN FINITE GROUPS

The unsolved problem of whether there exists a positive constant c such that the number k(G) of conjugacy classes in any finite group G satisfies k(G) >= c log(2) vertical bar G vertical bar has attracted attention for many years. Deriving bounds on k(G) from (that is, reducing the problem to) lower bounds on k(N) and k(G/N), N (sic) G, plays a critical role. Recently Keller proved the best lower bound known for solvable groups: k(G) > c(0) log(2)vertical bar G vertical bar/log(2)log(2)vertical bar G vertical bar (vertical bar G vertical bar >= 4) using such a reduction. We show that there are many reductions using k(G/N) >= beta [G : N](alpha) or k(G/N) >= beta(log[G : N])(t) which, together with other information about G and N or k(N), yield a logarithmic lower bound on k(G).