# prove H_T less than or equal to S_n

• Oct 13th 2008, 12:00 PM
mandy123
prove H_T less than or equal to S_n
Let T=[(a_1, a_2,...,a_t) C {1,2,...n}] and H_T={beta is a member of S_n|beta(a_i)=a_i, for all a_i which is a member of T}. H_t is called the stabilizer of T in S_n, stab(T). Prove that H_T < S_n. What is the order of H_T
• Oct 13th 2008, 12:16 PM
Jhevon
do you understand what H_T does?
Quote:

Originally Posted by mandy123
Let T=[(a_1, a_2,...,a_t) C {1,2,...n}] and H_T={beta is a member of S_n|beta(a_i)=a_i, for all a_i which is a member of T}. H_t is called the stabilizer of T in S_n, stab(T). Prove that H_T < S_n.

it suffices to show that H_T is a subgroup of S_n. or do you know that and are wondering how to do it?

you must show that H_T is nonempty. (consider the identity function)

then show that it is closed under the composition of functions

then show that each element has an inverse

which of these do you have problems with?

Quote:

What is the order of H_T
H_T leaves the elements in T fixed and permutes the other elements. how many ways can we permute the remaining elements?
• Oct 13th 2008, 05:41 PM
mandy123
ok to say it is nonempty:
beta(a_i)=a_i, so a_i^2=a_i So H is nonempty?

to show closure:
x,y is a member of H_t
beta(x)=a_i and beta(y)=a_i

Inverse: Dont know how to do this part

Did I get the other 2 right?
• Nov 16th 2008, 02:59 PM
Jhevon
Quote:

Originally Posted by mandy123
ok to say it is nonempty:
beta(a_i)=a_i, so a_i^2=a_i So H is nonempty?

to show closure:
x,y is a member of H_t
beta(x)=a_i and beta(y)=a_i

Inverse: Dont know how to do this part

Did I get the other 2 right?

note that the identity function is in $\displaystyle H_T$ so that it is not empty

to show closure, we need to show that if $\displaystyle \alpha (a_i) ,~\beta (a_i) \in H_T$ then (without loss of generality) $\displaystyle \beta \circ \alpha \in H_T$

so assume $\displaystyle \alpha , ~\beta \in H_T$. then $\displaystyle \alpha (a_i) = a_i$ and $\displaystyle \beta (a_i) = a_i$

thus $\displaystyle ( \beta \circ \alpha ) (a_i) = \beta ( \alpha (a_i)) = \beta (a_i) = a_i$

so that $\displaystyle \beta \circ \alpha \in H_T$ since $\displaystyle ( \beta \circ \alpha )(a_i) = a_i$

For inverses, note that if $\displaystyle \alpha \in H_T$ then $\displaystyle \alpha ^{-1} \in S_n$ works as the inverse of $\displaystyle \alpha$, and it is also in $\displaystyle H_T$, since if we assume $\displaystyle \alpha \in H_T$, then

$\displaystyle \alpha (a_i) = a_i$

$\displaystyle \Rightarrow \alpha ^{-1} (\alpha (a_i)) = \alpha ^{-1} (a_i)$

$\displaystyle \Rightarrow a_i = \alpha ^{-1}(a _ i)$

so that $\displaystyle \alpha ^{-1} \in H_T$

thus, $\displaystyle H_T \le S_n$