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17.17 (2010)

Solved

If a finitely generated group $G$ has $n < \infty$ maximal subgroups, must $G$ be finite? In particular, what if $n = 3$?

Contributor: G. M. Bergman

Progress

No, it need not. An example of an infinite group with 3 maximal subgroups is given by a 2-generated 2-LERF group in $\S\,7$ of (M. Ershov, A. Jaikin-Zapirain, J. Reine Angew. Math., 677 (2013), 71–134) as its maximal subgroups have index 2.

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