 So in the previous video we looked at what the stabilizer is of an element of a set when we act when we act with the set of a group on onto that element and I want to show you a tangible example and it's a type of a group that we haven't looked at before specifically but I'm going to just do that I'm going to take a square so square that side equals that side all the angles 90 degrees I'm going to call this corner one corner two corner three corner four with side one side two side three and side four and I'm going to have diagonal one and I'm going to have diagonal diagonal two oops diagonal two you see that and let's just think about if we fix certain points how can we rotate this thing and we're going to call row you know row is an rho row row zero is I do nothing to it and that's going to be our identity element row one is I stick a pin there and I wrote rotated anti-clockwise by 90 degrees so one goes to two three goes to four four you know it turns around like that so that is just the identity element I'm going to rotate counterclockwise by pi over two row two as I'm going to rotate by 180 degrees row three is three pi over two so 270 degrees mu one is mu one is what I'm going to do with mu one I'm going to flip it around a certain axis let's flip it around let's say for our example here I'm going to say let's let's just do that is vertical so I'm going to flip it around that axis so I keep it like that and I flip it around that axis and mu two is I'm going to flip it around this axis so I go up and I'm going to have d1 let's have d1 as I stick it there and I flip it around that axis and d2 I flip it around that axis so d1 d1 I drew this way d3 that way so I mean we can mix these two up but for now let's just imagine that that is our picture so I'm going to suggest that I have my group it's actually called d4 it's a group and it has these elements row zero row one row two row three mu one mu two and delta one and delta two and I have my binary operation is that I have this composition of two of these so first I rotate clockwise 90 degrees and then I flip around that axis so I'm not going to prove that for you Kelly's table that this is a group this is take for granted that it is a group and my set A is this going to be my sides my corners one two three four my sides side one side two side three side four and my diagonals one and my diagonal two so that's what I have now let's just look at the stabilizer of corner one what will which one of these elements in my group of the set that makes up my group will stabilize one that I can do any one of these and one will stay where it is definitely row zero row zero is going to be everywhere remember we showed we proved that this is a subgroup so it's got to be a group in its own right so the identity element is there what else will keep d1 in its place well if I if I flip along if I fix those two points and I just flip along there so along this line so d2 or I should not call that I should have called that delta 2 delta 1 and delta 2 so definitely flipping around delta 2 is going to do that for me and those are the only two under that and then under that binary operation as well so those two just keeping it on d2 and flipping it there is going to do that for me let's take side number three what is going to keep side number three there again the identity element but what about if I fix it along this axis here and I just flip it around now s3 turns around but s3 is still exactly where it is so definitely new one new one is going to be my other element and so I have an identity element it is those two are those two will be the inverses of which will be the inverses of each other so let's just look at well this one is its own inverse and that's its own inverse I should really say let's now look at a more complicated one let's keep diagonal one let's keep diagonal one in its place well first of all row will be there but what if I just were to keep keep it there and flip it around itself it will still be there so definitely definitely delta one will be there if I do this delta two and I flip it around there d will flip around but it is still just where it is so definitely delta two will be there and let's just think of of another one if if I just were to look at it and I were to rotate it through pi over two and another pi over two it stays there it's upside down but it's still there where it is so row two or row one there row two there row two there is also going to be so have a look at this beautiful example take take my word for it that that is a group and see if you agree with the stabilizes of we looked at that element we looked at side three and we looked at diagonal one see if you agree with what these stabilizes are of of those three elements of this set just to help you understand of cement what a stabilizer is