0:02 when you learn arithmetic you probably
0:04 had to memorize a multiplication table
0:06 this table shows you how to multiply any
0:09 two small numbers usually the integers 1
0:12 through 12 in abstract algebra you begin
0:15 to work with new types of numbers groups
0:17 behave very differently than the numbers
0:19 in arithmetic so when you're first
0:21 starting out an abstract algebra it's
0:23 helpful to go back to the basics and
0:26 make a group multiplication table group
0:27 multiplication tables are sometimes
0:30 called Kaye tables in of the British
0:33 mathematician Arthur Kaylee because he
0:34 was the first to use them in a math
0:36 paper they work just like the
0:38 multiplication tables from arithmetic
0:40 let's see a simple example and then look
0:42 at how Kaye tables can be used to
0:44 explore small
0:46 groups consider the group under
0:48 multiplication consisting of these four
0:52 elements 1 1 I and negative I it's
0:54 traditional to put the group operation
0:57 in the upper left corner next list the
0:59 elements in the same order in the header
1:00 row and header column colum starting
1:03 with the identity element one is the
1:05 identity element under multiplication so
1:07 any number multiplied by one is
1:10 unchanged this allows us to quickly fill
1:13 in the first row and First Column we can
1:14 now multiply and fill in the rest of the
1:23 table NE 1 * 1 is 1 -1 * I is I -1 * I
1:26 is I and so on I'd like to point out
1:28 some interesting features of this group
1:31 multiplication table one feature that
1:33 appears in every Kaye table is if you
1:35 start with the identity element then the
1:37 first row and First Column just repeat
1:39 the elements in the headers this is
1:41 because if you multiply any element by
1:44 the identity you get the same element
1:46 next every row and every column contains
1:48 the identity element why is this the
1:51 case this is because in a group every
1:53 element has an inverse in this group the
1:56 inverse of I is negative I because their
1:59 product is one another property worth
2:01 note is that this table is symmetric
2:04 about the diagonal if you flip the group
2:05 along the diagonal you get the same
2:09 table this is because the group is Aon a
2:14 * Bal B * a for any two elements A and B
2:16 if the group were non aelion then the
2:18 multiplication table would not be
2:20 symmetric there's another very important
2:22 property of this multiplication table
2:24 there are no duplicate elements in any
2:27 row or column we're not counting the
2:30 headers each row and each column contain
2:32 all the group elements in some order
2:34 let's prove that this happens for every
2:37 group suppose we replace this
2:39 multiplication table with one for an
2:41 arbitrary finite group let's assume
2:43 there's a row with duplicate elements
2:46 let's name the row a and the columns X
2:50 and Y then we have a * x = a * y if we
2:53 multiply on the left by the inverse of a
2:56 you get x equals y but X and Y are
2:58 different this contradiction shows our
3:01 assumption was wrong so there's no row
3:02 with duplicate
3:04 elements we can use the same reasoning
3:06 to show that there cannot be a column
3:08 with duplicate elements either if there
3:12 were then we'd get x * a = y * a
3:14 multiplying on the right by a inverse
3:18 gives us X = Y another
3:20 contradiction we're now going to use
3:23 Kaye tables to find the first few small
3:25 groups we'll begin with the simplest
3:28 case groups of order one quick reminder
3:30 the order of a group is just a number of
3:32 elements in the group and it's written
3:35 like this to start every group must have
3:38 an identity element we'll use e for the
3:41 identity Element e * e must equal e
3:42 since it's the only element around and
3:45 it's the identity element so there's a
3:47 single group of order one this cute
3:50 little group is called the trivial group
3:53 Next let's use multiplication tables to
3:55 find all groups of order two we'll call
3:57 the identity Element e and the second
4:00 element a since e is the identity
4:07 Element e * e = e e * a = a and a * E =
4:10 A this leaves only one empty Square in
4:13 the table we can now use the rule we
4:15 proved earlier every row and every
4:17 column must contain all the group
4:20 elements there are no duplicates since
4:22 the second row already contains a this
4:24 means the last Square must be e so
4:27 there's only one group of order two it
4:29 turns out that this is the same group as
4:32 the integers mod 2 try to convince
4:34 yourself of this by making the Kaye
4:35 table for the integers mod 2 under
4:39 addition then comparing the two
4:41 tables now let's find all groups of
4:43 order three we'll call the three
4:47 elements E A and B where e is the
4:49 identity element because e is the
4:51 identity element we can quickly fill in
4:53 the first row and First Column this
4:55 leaves four empty squares in our KV
4:58 table let's try to riddle out what goes
5:01 in this Square it can't be a because
5:03 this row already contains an a this
5:06 means it's either e or B let's guess e
5:10 and see what happens if a * a equal e
5:12 then the final Square in the second row
5:15 must be B this is because each row and
5:18 column has to contain all elements but
5:21 the moment we write B we see a problem
5:24 the third column now contains two B's
5:27 this is not allowed our guess was wrong
5:31 so a * a must be equal 2 b
5:33 we can now fill in the rest of the table
5:35 since we've already used A and B in row
5:39 two the last Square must be e similarly
5:41 since there is already an A and B in the
5:43 second column this empty Square must
5:46 also be e and lastly the third row
5:51 already has a b and an e so B * b equal
5:54 a so there is only one group of order
5:57 three and this is its Kaye table because
5:58 this table is symmetric about the
6:01 diagonal it's a an aelon group in fact
6:03 this group is identical to the integers
6:06 mod 3 under addition we say they are
6:09 isomorphic groups here's one way to see
6:11 this let's compare this table to the
6:14 Kaye table for the integers mod 3 if you
6:17 were to replace e with zero a with 1 and
6:20 B with two the two tables would be the
6:23 same by the way you may have noticed
6:25 that filling in Kaye tables is kind of
6:28 like solving a Sudoku puzzle each row
6:29 and each column must contain all the
6:32 elements of the group there can be no
6:34 duplicate elements keep this in mind for
6:36 the next challenge find all groups of
6:39 order four this one gets a bit trickier
6:42 but it's a very fun puzzle to solve here
6:44 are some Clues there are four possible
6:46 tables but three of them are actually
6:48 identical to each other so there are
6:50 only two groups of size four so grab
6:53 some pen and paper and give it a try and
6:55 help each other out in the comments
6:57 everyone has different sticking points
6:59 but we're all learning abstract algebra together
7:04 I'm Arthur kayy I say if I were around
7:06 today I would most definitely subscribe
7:08 to socratica not just because they
7:10 mentioned me in a video good chaps but
7:12 because our videos are most delightful
7:15 and what's this they're on patreon jolly
7:17 good cause jolly good a dollar here a
7:20 dollar there making loads more videos
7:23 yes yes jolly good cause indeed what is
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