Intersection of 3 Proper Subgroups

Welcome back to the world of magic Math tricks! Today we shall be covering an interesting problem that I found a few days ago on AoPS regarding the intersection of 3 proper subgroups (duh...). Apparently the problem was first proposed in the 9th number of 'Gazeta Matematica' from 1985. I surely do love problems that are older than me.

Let \(G\) be a group and \(H_1, H_2, H_3\) be proper subgroups of \(G\) such that \(G= H_1 \cup H_2 \cup H_3\)

Prove that \(x^2 \in H_1 \cap H_2 \cap H_3\) for all \(x \in G\).

Solution:

First of all, let us prove that none of this subgroups includes another. Let's presume by absurd that \(H_2\) is contained in \(H_1\). This would imply that \(H_1 \cup H_3 = G\), which, in turn, would imply that either \(H_1\) or \(H_3\) is \(G\), which contradicts the fact that the subgroups are proper. This argument works if one group is included in the reunion of the others, too.

If \(x \in H_1 \cap H_2 \cap H_3\), then it is obvious that \(x^2 \in H_1 \cap H_2 \cap H_3\).

If \(x \in H_1 \cap H_2 - H_3\), then we shall do a magic trick and write \(x^2=xaa^{-1}x\), where \(a \in H_3 - H_1 \cup H_2\). \(xa\) obviously doesn't belong to \(H_1 \cup H_2\), so it must be in \(H_3\). This argument works for \(a^{-1}x\), too, so \(x^2 \in H_1 \cap H_2 \cap H_3\). The other possible intersections of these groups can be treated the same.

If \(x \in H_1 - H_2 \cup H_3\), then we write \(x^2=xaa^{-1}x\) again, this time with \(a \in H_2 - H_1 \cup H_3\). Obviously, \(xa\) can't be in either \(H_2\) or \(H_1\), so it is in \(H_3\), so \(x^2 \in H_3\). By choosing a suitable \(b \in H_3 - H_1 \cup H_2\), we get that \(x^2 \in H_2\), so \(x^2 \in H_1 \cap H_2 \cap H_3\). All similar cases are treated as above.

This little exercise can work as a Mathematical 'Creativity Test' if you think about it, because it's about discerning a certain pattern and coming up with an idea that solves the problem instantly. More posts might be coming this week, as I am working on 2 more articles atm: the psychology article (finally) and another one with partial derivatives / Lagrange multipliers. Tune in next time for one of these articles (probably the psychology article will come first)!

Cris.