Abstract
We prove first that if G is a finite solvable group of derived length d ≥ 2, then k(G) > |G|1/(2d−1), where k(G) is the number of conjugacy classes in G. Next, a growth assumption on the sequence [G(i): G(i+1)] d−11 , where G(i) is theith derived group, leads to a |G|1/(2d−1) lower bound for k(G), from which we derive a |G|c/log 2log2|G| lower bound, independent of d(G). Finally, “almost logarithmic” lower bounds are found for solvable groups with a nilpotent maximal subgroup, and for all Frobenius groups, solvable or not.
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Bertram, E.A. Lower bounds for the number of conjugacy classes in finite solvable groups. Israel J. Math. 75, 243–255 (1991). https://doi.org/10.1007/BF02776026
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DOI: https://doi.org/10.1007/BF02776026