Is there an infinite finitely generated group whose Frattini factor is isomorphic to Klein 4-group?
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asked Feb 20, 2016 at 13:03
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$\begingroup$ By Frattini factor, do you mean its Frattini subgroup or the quotient by the Frattini subgroup? Or something else? $\endgroup$– HJRWCommented Feb 20, 2016 at 13:35
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$\begingroup$ @HJRW. I mean the quotient by the Frattini subgroup. $\endgroup$– Alireza AbdollahiCommented Feb 20, 2016 at 13:39
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$\begingroup$ 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). $\endgroup$– Geoff RobinsonCommented Feb 25, 2016 at 0:54
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$\begingroup$ @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). $\endgroup$– Alireza AbdollahiCommented Feb 26, 2016 at 19:12
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$\begingroup$ @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. $\endgroup$– Alireza AbdollahiCommented Jan 14, 2017 at 13:13
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