YouTube Transcript: Group Multiplication Tables | Cayley Tables (Abstract Algebra)
Skip watching entire videos - get the full transcript, search for keywords, and copy with one click.
Share:
Video Transcript
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 in abstract algebra you begin
to work with new types of numbers groups
behave very differently than the numbers
in arithmetic so when you're first
starting out an abstract algebra it's
helpful to go back to the basics and
make a group multiplication table group
multiplication tables are sometimes
called Kaye tables in of the British
mathematician Arthur Kaylee 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 Kaye tables can be used to
explore small
groups consider the group under
multiplication consisting of these four
elements 1 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 colum starting
with the identity element one is the
identity element under multiplication so
any number multiplied by one 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 NE 1 * 1 is 1 -1 * I is I -1 * 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 Kaye 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 one another property worth
note 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 Aon a
* Bal B * a for any two elements A and B
if the group were non aelion 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 one 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 * x = a * 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 * a = y * a
multiplying on the right by a inverse
gives us X = Y another
contradiction we're now going to use
Kaye tables to find the first few small
groups we'll begin with the simplest
case groups of order one 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 * 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 one this cute
little group is called the trivial group
Next let's use multiplication tables to
find all groups of order two we'll call
the identity Element e and the second
element a since e is the identity
Element e * e = e e * a = a and a * E =
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 order two it
turns out that this is the same group as
the integers mod 2 try to convince
yourself of this by making the Kaye
table for the integers mod 2 under
addition then comparing the two
tables now let's find all groups of
order three 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
leaves four empty squares in our KV
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 e
and see what happens if a * a equal 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 * a must be equal 2 b
we can now fill in the rest of the table
since we've already used A and B in row
two 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 an e so B * b equal
a so there is only one group of order
three and this is its Kaye table because
this table is symmetric about the
diagonal it's a an aelon 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
Kaye table for the integers mod 3 if you
were to replace e with zero a with 1 and
B with two the two tables would be the
same by the way you may have noticed
that filling in Kaye 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 four 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 four 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 kayy I say if I were around
today I would most definitely subscribe
to socratica not just because they
mentioned me in a video good chaps but
because our videos are most delightful
and what's this they're on patreon jolly
good cause jolly good a dollar here a
dollar there making loads more videos
yes yes jolly good cause indeed what is
Click on any text or timestamp to jump to that moment in the video
Share:
Most transcripts ready in under 5 seconds
One-Click Copy125+ LanguagesSearch ContentJump to Timestamps
Paste YouTube URL
Enter any YouTube video link to get the full transcript
Transcript Extraction Form
Most transcripts ready in under 5 seconds
Get Our Chrome Extension
Get transcripts instantly without leaving YouTube. Install our Chrome extension for one-click access to any video's transcript directly on the watch page.
Chrome Extension