Arthur Cayley
Last week we learned about Sylvester, this week let's look at one of his dearest friends who also had a liking of mathematics.
Sylvester commented, "Mr. Cayley, of whom it may be so truly said, whether the matter he takes in hand be great or small, "nihil tetigit quod non ornavit," which translates to "He touched nothing without embellishing it". Indeed Cayley had a knack for mathematics and many things. Born on August 16, 1821 in England, Cayley ended up writing enough papers on mathematics to create thirteen volumes, about 600 pages each, of just his work. He also had an interest in astronomy, mechanics, painting, traveling, architecture, and hiking. He taught at John Hopinks University for a bit and was a lawyer for fourteen years- the whole time writing papers for mathematics. Eventually he became a professor of mathematics at Cambridge University (also getting married the same year) until he died on January 26, 1895. There he not only taught mathematics and carried out research, but he successfully pushed for women to be accepted into Cambridge as well.
Due to Cayley's amount of work mentioned above, it is not hard to believe that the man has contributed to many fields of mathematics- he helped come up with the theory of invariants, he worked with linear algebra, geometry, he invented Cayley graphs, Cayley tables, and worked quite a bit more with groups, the list could go on. For right now let's focus specifically on groups and Cayley tables as Abstract Algebra is all about groups. This will be fun!
Imagine you just got a brand new car (Santa was extra giving this year...).
It is in beautiful condition, but in awhile will need new tires. You buy tire 1, tire 2, tire 3, and tire 4. There is no specific order you have to put the tires on even though they are numbered. How many different combinations can you come up with? (*You should be able to put together 8 different combinations!)
While doing this, you may notice that you cannot move just one tire. If you want to switch one tire that means you are going to have to move at least one other tire. One other option is to move nothing and leave everything in its original place.
Now that you have thought about this for a bit, let's look at something very similar with just a plain square.
Here each vertex is labeled, we have a vertex set! Let's call our new set $V$ and let's say that $V =$ {$1, 2, 3, 4$}. We also have a set of edges, call this set $E$ and say our set $E =$ {{$1,2$}, {$2,3$}, {$3,4$}, {$4,1$}}. We now have a graph where $G = (V, E)$. How many different ways can we move these vertices? Well, squares are made from right angles and right angles are $90^{\circ}$right? So let's just rotate this square clockwise a couple times and see what we get.
$R_{0}$ gives, $R_{90}$ gives, $R_{180}$ gives, $R_{270}$ gives,
$H$ gives, $V$ gives, $D$ gives, $D'$ gives,
The top row is what we should get when shifting everything clockwise $90^{\circ}$ several times and includes what happens when we leave everything in its original spot, like what we mentioned earlier. The bottom row has first a horizontal flip, a vertical flip, a diagonal flip and then an off diagonal flip.
These eight different movements make up a specific group called $D_4$. And actually each movement that makes up $D_4$ is considered a function. So what happens when we combine two of these movements? Is that allowed? Well, when we think of these movements as functions we can then use function composition. Function composition works like so, let $f$ and $g$ be functions so then $f \circ g$ means "$g$ followed by $f$". Using our combination of movements let's apply this same idea. Let's take $HR_{90}$ which means "$R_{90}$ followed by $H$". This new combination of movements will give us $HR_{90} = D$. We can make more movements by combining our original eight from $D_4$! So then how many combinations of movements can we make? We can actually put all of this into a table called a Cayley table believe it or not, named after Arthur Cayley, the man who invented it.
And now you have explored a small part of Arthur Cayley's work with groups and gained a basic understanding of what we look at in Abstract Algebra. If you are still interested in groups and Cayley Tables, here is a link to a cite that goes a little more in depth on the subject.
Thanks bunches,
-Haylee Jo Lau
(P.S. Here is a cool video my Abstract Algebra teacher shared with our class to get us excited about groups, hope you enjoy!)Works Cited:
Contemporary Abstract Algebra, Joseph A. Gallian, Ninth Edition
That's awesome never new about any of this!
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