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.