# Thread: find the centre of S_n

1. ## find the centre of S_n

question 1) Prove that $\displaystyle Z(S_n)=1 \text{ for all } n \geq 3$.

question 2) Prove that if $\displaystyle H \trianglelefteq G$ with $\displaystyle |G:H|=p,\text{ p is a prime}$, then for all subgroups K of G either $\displaystyle K \leq H$ or,
$\displaystyle G=HK \text{ and } |K:K \cap H|=p$.

I dont know how to solve the first one but i could solve the second one.
here is my solution of the second one:

since $\displaystyle H \trianglelefteq G$ we find that $\displaystyle HK = KH$ so $\displaystyle HK \leq G$.
also $\displaystyle H \trianglelefteq HK$ and $\displaystyle HK/H \leq G/H$ so it makes sense to consider $\displaystyle |G/H:HK/H|$.

Using $\displaystyle |HK|= (|H||K|)/|H \cap K|$ we get,

$\displaystyle |G/H:HK/H| = |G:H||H|/|HK| = p|H \cap K|/|K| = p/|K: H \cap K|$.

This means either $\displaystyle |K: H \cap K|=1 \text{ or } p$
this easily lads to the desired result.

If you have a different solution to the second one then please post it.

2. Originally Posted by abhishekkgp
question 1) Prove that $\displaystyle Z(S_n)=1 \text{ for all } n \geq 3$.

question 2) Prove that if $\displaystyle H \trianglelefteq G$ with $\displaystyle |G:H|=p,\text{ p is a prime}$, then for all subgroups K of G either $\displaystyle K \leq H$ or,
$\displaystyle G=HK \text{ and } |K:K \cap H|=p$.

I dont know how to solve the first one but i could solve the second one.
here is my solution of the second one:

since $\displaystyle H \trianglelefteq G$ we find that $\displaystyle HK = KH$ so $\displaystyle HK \leq G$.
also $\displaystyle H \trianglelefteq HK$ and $\displaystyle HK/H \leq G/H$ so it makes sense to consider $\displaystyle |G/H:HK/H|$.

Using $\displaystyle |HK|= (|H||K|)/|H \cap K|$ we get,

$\displaystyle |G/H:HK/H| = |G:H||H|/|HK| = p|H \cap K|/|K| = p/|K: H \cap K|$.

This means either $\displaystyle |K: H \cap K|=1 \text{ or } p$
this easily lads to the desired result.

If you have a different solution to the second one then please post it.
For the first one....don't think too hard. Given any element of $\displaystyle S_n\;\; n\geqslant 3$ just construct something which doesn't commute with it. There are two more ways I can see to prove the second one, but the one you used is easiest.

3. Originally Posted by Drexel28
For the first one....don't think too hard. Given any element of $\displaystyle S_n\;\; n\geqslant 3$ just construct something which doesn't commute with it. There are two more ways I can see to prove the second one, but the one you used is easiest.
i got the solution to the first one after i posted it.
can you post your solutions of the second one???