I was interested in knowing if groups with following property have been studied( like what can be said about structure of the group) : "$G$ can be written as disjoint union of a given number of abelian proper subgroups". (this number is not necessarily smallest such number)
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2$\begingroup$ Interpreting disjoint as "any two intersect only in the identity",such a group is itself Abelian, as any two such normal subgroups would centralize each other. $\endgroup$– Geoff RobinsonCommented Apr 19, 2016 at 7:29
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1$\begingroup$ I am sorry, I had just realized that I did not want normal and now I see why :) @GeoffRobinson $\endgroup$– user101Commented Apr 19, 2016 at 8:11
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$\begingroup$ See groupprops.subwiki.org/wiki/…. $\endgroup$– Stefan Kohl ♦Commented Apr 21, 2016 at 16:39
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$\begingroup$ @StefanKohl That article has no substantive content beyond the definition. $\endgroup$– zibadawa timmyCommented Apr 24, 2016 at 3:57
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2 Answers
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A result of mine (appearing as Problem 2.10(b) of my character theory book) says that if $G$ is nonabelian and is a disjoint union of $n$ abelian subgroups, then each of these subgroups has order at most $n-1$ and $|G|$ is at most $(n-1)^2$. As far as I know, there is no non-character proof of this result (though I have not tried very hard to find one).
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In the case when the nonabelian $p$-group $G$ admits a partition by cyclic subgroups, an answer is known: either $\exp(G)=p$ or the Hughes subgroup of $G$ is a proper subgroup of $G$. In the general case, when a $p$-group $G$ admits a non-trivial partition, the answer is also known (M. Suzuki). There are papers of R. Baer, O. Kegel, M. Suzuki and other authors devoted to finite groups with partition (see Mathscinet).
