Group Multiplication Tables | Cayley Tables (Abstract Algebra)

when you learn arithmetic you probably had to memorize a multiplication table this table shows you how to multiply any two small numbers usually the integers 1 through 12 an abstract algebra you begin to work with new types of numbers groups behave very differently than the numbers and arithmetic so when you're first starting out in abstract algebra it's helpful to go back to the basics and make a group multiplication table group multiplication tables are sometimes called Cayley tables in honor of the British mathematician Arthur Cayley because he was the first to use them in a math paper they work just like the multiplication tables from arithmetic let's see a simple example and then look at how Cayley tables can be used to explore small groups consider the group under multiplication consisting of these 4 elements 1 negative 1 I and negative I it's traditional to put the group operation in the upper left corner next list the elements in the same order in the header row and header column starting with the identity element 1 is the identity element under multiplication so any number multiplied by 1 is unchanged this allows us to quickly fill in the first row and first column we can now multiply and fill in the rest of the table negative 1 times negative 1 is 1 negative 1 times I is negative I negative 1 times negative I is I and so on I'd like to point out some interesting features of this group multiplication table one feature that appears in every Cayley table is if you start with the identity element then the first row and first column just repeat the elements in the headers this is because if you multiply any element by the identity you get the same element next every row and every column contains the identity element why is this the case this is because in a group every element has an inverse in this group the inverse of I is negative I because their product is 1 another property worth noting is that this table is symmetric about the diagonal if you flip the group along the diagonal you get the same table this is because the group is abelian a times B equals B times a for any two elements a be if the group were non-abelian then the multiplication table would not be symmetric there's another very important property of this multiplication table there are no duplicate elements in any row or column we're not counting the headers each row and each column contain all the group elements in some order let's prove that this happens for every group suppose we replace this multiplication table with 1 for an arbitrary finite group let's assume there's a row with duplicate elements let's name the row a and the columns x and y then we have a times X equals a times y if we multiply on the left by the inverse of a you get x equals y but x and y are different this contradiction shows our assumption was wrong so there's no row with duplicate elements we can use the same reasoning to show that there cannot be a column with duplicate elements either if there were then we'd get x times a equals y times a multiplying on the right by a inverse gives us x equals y another contradiction we're now going to use Cayley tables to find the first few small groups we'll begin with the simplest case groups of order 1 quick reminder the order of a group is just a number of elements in the group and it's written like this to start every group must have an identity element we'll use e for the identity element e times e must equal e since it's the only element around and it's the identity element so there's a single group of order 1 this cute little group is called the trivial group next let's use multiplication tables to find all groups of order 2 we'll call the identity element e and the second element a since E is the identity element e times e equals e e times a equals a and a times e equals a this leaves only one empty square in the table we can now use the rule we proved earlier every row and every column must contain all the group elements there are no duplicates since the second row already contains a this means the last square must be e so there's only one group of or - it turns out that this is the same group as the integers mod to try to convince yourself of this by making the Cayley table for the integers mod 2 under addition then comparing the two tables now let's find all groups of order 3 we'll call the three elements e a and B where E is the identity element because E is the identity element we can quickly fill in the first row and first column this Li is four empty squares in our Cavey table let's try to riddle out what goes in this square it can't be a because this row already contains an A this means it's either E or B let's guess eight and see what happens if a times a equals E then the final square in the second row must be B this is because each row and column has to contain all elements but the moment we write B we see a problem the third column now contains two B's this is not allowed our guess was wrong so a times a must be equal to B we can now fill in the rest of the table since we've already used a and B in Row 2 the last square must be e similarly since there is already an A and B in the second column this empty square must also be e and lastly the third row already has a B and any so B times B equals a so there is only one group of order 3 and this is its Cayley table because this table is symmetric about the diagonal it's an abelian group in fact this group is identical to the integers mod 3 under addition we say they are isomorphic groups here's one way to see this let's compare this table to the Cayley table for the integers mod 3 if you were to replace e with 0 a with 1 and B with two the two tables would be the same by the way you may have noticed that filling in Keighley tables is kind of like solving a Sudoku puzzle each row and each column must contain all the elements of the group there can be no duplicate elements keep this in mind for the next challenge find all groups of order 4 this one gets a bit trickier but it's a very fun puzzle to solve here are some clues there are four possible tables but three of them are actually identical to each other so there are only two groups of size 4 so grab some pen and paper and give it a try and help each other out in the comments everyone has different sticking points but we're all learning abstract algebra together I'm Arthur Cayley I say if I were around today I would most definitely subscribe to Socratic ax not just because they mentioned me in a video good chaps but because of videos are most alike --fill and what's this they're on patreon charlie good cars charlie good a dollar here dollar there making loads more videos yes yes jolly good cause indeed Oh what is that a shrubbery

Channel: Socratica

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Keywords: Socratica, SocraticaMath, abstract algebra, group, groups, math, maths, mathematics, Arthur Cayley, Cayley table, Cayley tables, group multiplication, group multiplication table, STEM, modern algebra, college, university, college math, higher math, advanced math, tutor, educational video

Id: BwHspSCXFNM

Length: 7min 31sec (451 seconds)

Published: Sun Sep 11 2016