## Universal Property of Kernel Question

Recently, the Dayan Zhanchi brand Rubik’s Cube has been a hot sale on my site. Parents looking to buy an educational toy may want to check it out. Smoothness when turning is very important for Rubik’s Cube, and the Dayan Zhanchi is one of the smoothest cubes out there (much better than the ‘official’ brand).

In this blog post, we will discuss a category theory question, in the framework of homomorphisms of abelian groups.

Let $\phi:M'\to M$ be a homomorphism of abelian groups. Suppose that $\alpha:L\to M'$ is a homomorphism of abelian groups such that $\phi\circ\alpha$ is the zero map. (One example is the inclusion $\mu:\ker\phi\to M'$)

Are the following true or false?

(i) There is a unique homomorphism $\alpha_0:\ker\phi\to L$ such that $\mu=\alpha\circ\alpha_0$.

(ii) There is a unique homomorphism $\alpha_1:L\to\ker\phi$ such that $\alpha=\mu\circ\alpha_1$.

It turns out that (i) is false. We may construct a trivial counterexample as follows. Consider $L=M=0$, and $M'=\mathbb{Z}/2\mathbb{Z}$. Let $\alpha$, $\phi$ be both the zero maps. Then certainly $\phi\circ\alpha=0$. $\ker\phi=\mathbb{Z}/2\mathbb{Z}$. Then, for any $\alpha_0$, $\alpha\circ\alpha_0(x)=0$, and hence is not equals to the the inclusion map $\mu$.

It turns out that (ii) is true, in fact it is the famous universal property of the kernel, that any homomorphism yielding zero when composed with $\phi$ has to factor through $\ker\phi$.

First we will prove uniqueness. Let $\alpha=\mu\circ\alpha_1=\mu\circ\beta$, where $\beta$ is another such map with the property (ii). Then for all $x\in L$, $\mu\alpha_1(x)=\mu\beta(x)$, which implies $\mu(\alpha_1(x)-\beta(x))=0$. Since $\mu$ is the inclusion map, this means that $\alpha_1(x)-\beta(x)=0$ and thus $\alpha_1(x)=\beta (x)$.

Next, we will prove existence. Consider $\alpha_1:L\to\ker\phi, \alpha_1(l)=\alpha(l)$. Note that $\phi(\alpha(l))=0$ by definition thus $\alpha(l)\in\ker\phi$.

Next we prove it is a homomorphism. $\alpha_1(l_1l_2)=\alpha(l_1l_2)=\alpha(l_1)\alpha(l_2)=\alpha_1(l_1)\alpha_1(l_2)$.

Finally by construction it is easy to see that $\mu\alpha_1(l)=\mu\alpha(l)=\alpha(l)$ for all $l\in L$.

## Best Rubik’s Cube (Cheap and Good) — The Dayan Zhanchi?

The Rubik’s Cube is a famous puzzle, that is related to Math and Group Theory. (See this free introduction by MIT on the The Mathematics of the Rubik’s Cube)

Recently, I am thinking of buying a new Rubik’s Cube, and searched on the internet on what is the best brand of Rubik’s Cube. For Rubik’s Cube, smoothness while turning is really important, because it will simply be easier to turn the edges if the cube is smooth.

After researching online, I came to a very surprising conclusion: The “made in China” brand Dayan Zhanchi is supposedly much better than the official Rubik’s brand (and also other “Western” brands)!

This amazing superhuman World Record is set using the Dayan Zhanchi! (2013, Mats Valk)

Other than the Dayan series, another alternative is the V-cube series:

V-CUBE 3 White Multicolor Cube

However, the reviews on Amazon seem to indicate that the Dayan is superior in both smoothness and price!

If you have any recommendations on which Rubik’s Cube is best, please write in the comments below!

I will be buying the Dayan Cube soon (hopefully in time for Christmas 2014), and will post new updates! I am most probably buying the stickerless version since I have past experience of stickers falling off from my previous cubes. (Note: Stickerless Rubik’s Cubes are banned from competitions for the ridiculous reason that it is possible to “see what colors are behind through the cracks”, see https://github.com/cubing/wca-documents/issues/177) So if your goal is to enter a competition, you may want to consider the sticker version of the Zhanchi.

For parents, buying a Rubik’s cube for your child is a great investment. Playing with the Rubik’s cube is a major intellectual challenge (it has 43 quintillion permutations, only 1 of which is correct), which will develop the child’s brain for logical thinking, which is especially useful for Math and Science. Most importantly, it is fun!

# Special note for buying Dayan Zhanchi from Singapore:

If you are buying the Dayan Zhanchi from Singapore, at first it seems like the Dayan ZhanChi does not ship to Singapore. It actually does! We just have to choose the correct seller, Cube Puzl, which ships to Singapore.