Following Tomkinson (1975), I define the Frattini-like subgroup $\mu(G)$ as the intersection of all major subgroups of $G$. Is there a program or procedure one can use to determine (or construct) all the major subgroups of a group $G$ and hence identify $\mu(G)$?
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6$\begingroup$ Could you tell us what the definition of 'major subgroup' is? $\endgroup$– Colin ReidCommented Nov 15, 2011 at 14:04
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2$\begingroup$ Here is the definition: subgroup $M$ of a group $G$ is called major if $M \le U < G$ implies that the least upper bounds for the transfinite length of properly ascending chains between $M$ and $G$ and between $Y$ and $G$ are the same. In particular, if $G$ is finite, major subgroups = maximal subgroups. $\endgroup$– user6976Commented Nov 15, 2011 at 16:34
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For finite groups $\mu(G)$ is the Frattini subgroup (see Tomkinson, M. J. A Frattini-like subgroup. Math. Proc. Cambridge Philos. Soc. 77 (1975), 247–257. ). For infinite groups, even finitely presented, there can't be any algorithm since most natural algorithmic problems are undecidable in general. As I understand, this concept is studied mostly for locally finite (or at least locally nice) infinitely generated groups. In that case even the notion of "algorithm" is not well defined. I guess you just wanted some info about how to deal with major subgroups. The latest article about them that I could find was: Walter, Vonn, Finiteness conditions and a Frattini-like subgroup in locally nilpotent groups. Ricerche Mat. 54 (2005), no. 1, 39–44 (2006).