Anyone recognize this subgroup?

[Juanriq]

Ahoy hoy, let A be a set with $a \in A$. Define

$G_a = \{ g \in S_A; g(a) = a \}$

Where $S_A$ is the permutation group. Are we just talking the set of all inverses of the permutation group? Thanks!

 

[lanedance]

isn't it the set which leaves a unchanged in a permutation?

 

[micromass]

Have you seen the concept of "actions" yet?

 

[Juanriq]

Hey guys, I think I don't understand what this subgroup does. Lanedance, I think I see what you mean, but wouldn't that just be like an identity group? I don't understand what would be so special about some group that just spits the same thing back out all the time... Also, micromass, I have not seen actions yet. This came up in a geometry class after we spent 5 minutes on the permutation group. I understand what the permutation group is, but this subgroup really means nothing to me. Any enlightening ideas would be welcome, I'm going to go wikipedia actions.

 

[lanedance]

Hey guys, I think I don't understand what this subgroup does. Lanedance, I think I see what you mean, but wouldn't that just be like an identity group? I don't understand what would be so special about some group that just spits the same thing back out all the time...

as i read it, it only has to leave the element a unchanged

 

[Juanriq]

so if a=(321), then g(321) = (321), right?

 

[Juanriq]

the inverse of $G_a$ is the identity?

 

[lanedance]

i would have thought it was somthing like
A = {a,b,c}

the set of all permutations of A is $S_A$
{a,b,c} ()
{b,a,c} (12)
{c,b,a} (13)
{a,c,b} (23)
{b,c,a} (123)

the set of all permutations that leave a unchanged are $S_a$
{a,b,c} ()
{a,c,b} (23)

 

[Juanriq]

Ohhhhh... that's something completely different than what I was thinking. Thanks lanedance!