Let $G=\langle a,b\rangle$ be a finite supersolvable group. Is there any special information about the structure of $G$ when $o(a)=2$ and $ o(b)=2^k > 2$?
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$\begingroup$ Usually, two-generators groups are well-suited for being represented on a surface. You may look for keywords like "hyperbolic groups" or "Fuchsian group". You use the Cayley graph and the natural action of the group on it (see, e.g., sciencedirect.com/science/article/pii/0095895683900096 ). In the most cases, the group is infinite (see the Burnside problem). $\endgroup$– Giovanni MorenoApr 16, 2016 at 6:09
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