8.10 (1982)
SolvedIs the group $G = \langle a, b \mid a^n = 1, ab = b^3 a^3 \rangle$ finite or infinite for $n = 7$, $n = 9$, and $n = 15$? All other cases known. See also 7.7.
Progress
Infinite in all three cases. For $n = 15$ by (D. J. Seal, Proc. Roy. Soc. Edinburgh (A), 92 (1982), 181–192). For $n = 9$ (and 15) by (M. I. Prishchepov, Commun. Algebra, 23 (1995), 5095–5117). For $n = 7$, because it contains the Fibonacci group $F(3, 7)$ as an index 7 subgroup, as follows from Theorem 3.0 of (C. P. Chalk, Commun. Algebra, 26, no. 5 (1998), 1511–1546) by standard technique for working with Fibonacci groups (G. Williams, Letter of 6 October 2015).
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