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Jan 14, 2017 at 13:20 comment added Alireza Abdollahi @AndreiJaikin I wish he can put his answer himself here! This answers (Problem 17.17 of THE KOUROVKA NOTEBOOK, No. 18; arxiv.org/pdf/1401.0300v8.pdf) proposed by G.M.Bergman: If a finitely generated group $G$ has $n < \infty$ maximal subgroups, must $G$ be finite? In particular, what if $n = 3$?
Jan 14, 2017 at 13:13 comment added Alireza Abdollahi @AndreiJaikin. Andrei Jaikin-Zapirain has sent an answer in Group-Pub-forum at 2016/Aug/21 as follows: An example of an infinite group with 3 maximal subgroups can be extracted from the paper Ershov, Mikhail; Jaikin-Zapirain, Andrei Groups of positive weighted deficiency and their applications. J. Reine Angew. Math. 677 (2013), 71–134. A 2-generated 2-LERF group (see Section 7 of the paper) has this property because its maximal subgroups have index 2.
Feb 26, 2016 at 19:12 comment added Alireza Abdollahi @GeoffRobinson. Actually it is equivalent to ask if there is an infinite finitely generated group with exactly 3 maximal subgroups (without any other restriction on the indices).
Feb 25, 2016 at 0:54 comment added Geoff Robinson I think an equivalent form of the question is whether there can be an infinite group $G$ which has exactly three maximal subgroups each of index $2$ ( if there is such a $G$, it must be generated by two elements).
Feb 20, 2016 at 13:41 review Low quality posts
Feb 20, 2016 at 13:53
Feb 20, 2016 at 13:41 history edited Alireza Abdollahi CC BY-SA 3.0
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Feb 20, 2016 at 13:39 comment added Alireza Abdollahi @HJRW. I mean the quotient by the Frattini subgroup.
Feb 20, 2016 at 13:35 comment added HJRW By Frattini factor, do you mean its Frattini subgroup or the quotient by the Frattini subgroup? Or something else?
Feb 20, 2016 at 13:21 review Low quality posts
Feb 20, 2016 at 13:24
Feb 20, 2016 at 13:03 history asked Alireza Abdollahi CC BY-SA 3.0