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Stabilisers and fixed set

Hi, just need to see if my answers are correct for the below as I keep getting confused with this topic.

Question

The following equation defines a group action of (R*, x) on the plane R^2

C ^ (x,y) = (cx,y) ^ is 'wedge'

Determine the stabilisers of (1,0) and (0,1)

Also, find the fixed set Fix(3).

Answer

stab(1,0) = R*
stab (0,1) = R*

Fix(3) = (3x,y) = (0,y) the y-axis
Reply 1
Avatar for Mark13
Mark13
Original post by jojo55
Hi, just need to see if my answers are correct for the below as I keep getting confused with this topic.

Question

The following equation defines a group action of (R*, x) on the plane R^2

C ^ (x,y) = (cx,y) ^ is 'wedge'

Determine the stabilisers of (1,0) and (0,1)

Also, find the fixed set Fix(3).

Answer

stab(1,0) = R*
stab (0,1) = R*

Fix(3) = (3x,y) = (0,y) the y-axis


I don't agree with your first answer - think again about which elements of R* fix (1,0). The last two answers look good to me though.
Reply 2
Avatar for jojo55
jojo55
OP
11
Original post by Mark13
I don't agree with your first answer - think again about which elements of R* fix (1,0). The last two answers look good to me though.


Thanks for clarifying. I'm not too sure still about the first one then but I'll keep at it.
Reply 3
Avatar for jojo55
jojo55
OP
11
Original post by Mark13
I don't agree with your first answer - think again about which elements of R* fix (1,0). The last two answers look good to me though.


is it {1}?
Reply 4
Avatar for Mark13
Mark13
Original post by jojo55
is it {1}?


Yep
Reply 5
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jojo55
OP
11
Original post by Mark13
Yep


Woo! :biggrin:

Thanks
Reply 6
Avatar for Mark85
Mark85
16
Original post by jojo55


Fix(3) = (3x,y) = (0,y) the y-axis


Just to add to what has already been said, what you have written here makes no sense:

Fix(3) is the set of points in the plane (x,y) such that 3 ^ (x,y) = (x,y).

So Fix(3) isn't equal to (3x,y) it is equal to the set of (x,y) such that (3x,y)=(x,y) which is equal to the set of all points (0,y).

This might seem pedantic but the more you write things that don't make any grammatical sense when you try to answer questions - the more you will just confuse yourself in future. It doesn't take much longer to write

Fix(3) = {(x,y) e R^2 |3x=x} = {(0,y)|y e R^2}

than what you wrote but at least it requires you to think about what type of objects you are dealing with.
Reply 7
Avatar for jojo55
jojo55
OP
11
Original post by Mark85
Just to add to what has already been said, what you have written here makes no sense:

Fix(3) is the set of points in the plane (x,y) such that 3 ^ (x,y) = (x,y).

So Fix(3) isn't equal to (3x,y) it is equal to the set of (x,y) such that (3x,y)=(x,y) which is equal to the set of all points (0,y).

This might seem pedantic but the more you write things that don't make any grammatical sense when you try to answer questions - the more you will just confuse yourself in future. It doesn't take much longer to write

Fix(3) = {(x,y) e R^2 |3x=x} = {(0,y)|y e R^2}

than what you wrote but at least it requires you to think about what type of objects you are dealing with.


Thanks. I get confused when trying to explain it as I don't fully understand it myself.

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