# gr.group theory – Breuer-Guralnick-Kantor conjecture and infinite generated groups in 3/2

A group $$G$$ is called $$frac {3} {2}$$-generated if each non-trivial element is contained in a generating pair, that is to say $$rangle = G.$$

See this beautiful Scott Harper poster.
Proposal: Yes $$G$$ is $$frac {3} {2}$$-generated then each own quotient of $$G$$ is cyclical (proof).
Conjecture (B.G.K.): A finite group is $$frac {3} {2}$$-if generated if each correct quotient is cyclical.
Theorem (G.K.): Each simple finite group is $$frac {3} {2}$$-généré.

Question: Can the conjecture above be extended to finite groups?
In other words: is there a counterexample (known) for such groups?

Of course, any simple group that is not finished (like the alternating infinite group) $$A _ { infty}$$) has all appropriate cyclic quotients, but is not $$frac {3} {2}$$-généré.