00:01
Okay so we're given two permutations there is a little bit of a definition question but here's the thing we want to find out f o g and g o f so f o g says apply take our one two three four apply g first and then f all right so put in our top row so g says take one and turn it into three then f says take three that also gives us three back g says take two turn it into one and then f says take one and turn it into two g says take three turn it into four and then f says take four and turn it back into four and then g g says four goes to two, and then f says two goes to one.
01:08
So there's f -o -g.
01:12
And then g -o -f, we do it the other way around.
01:17
We do f first, and then g.
01:24
So f says one goes to two, and then g says two goes to one.
01:32
And then f says two goes to one, g says one goes to three.
01:37
And then f says 3 goes to 3 and g says 3 goes to 4 and then here we got 4 to 4 and then 4 to 2 so there's gof we can see that they're not the same so in other words it's non -abelian so the so the fog and fgof are not the same so they don't commute okay so f inverse well well, so f inverse is the thing that does the opposite.
02:26
All right.
02:27
So, in fact, the thing about f inverse, it turns out, what we do is we start at the bottom row and go up.
02:45
So 1 goes to 2 and 2 goes to 1 and then 3 goes to 3 and 4 goes to 4.
02:58
So we read that off by starting with numbers in the bottom row and finding out what they came from in the top row.
03:10
Okay.
03:15
And then it's easy to see that fof inverse and f inverse of is equal to the identity element where everything goes into the same thing.
03:36
That's pretty obvious.
03:38
And in fact, in this particular case, f equals f inverse.
03:46
That doesn't often happen, but it does happen.
03:53
And then we're supposed to, in part c, find g inverse.
03:58
So remember what we do.
03:59
We take, all right, and then we'll start at the bottom row.
04:08
So in g, 1 goes to 2, going backward.
04:17
And then 2 goes to 4...