Prove HK=KH iff HK is a Subgroup of G

[annoymage]

Homework Statement

if H and K are arbitrary subgroup of G, prove that HK=KH iff HK is a subgroup of G

Homework Equations

n/a

The Attempt at a Solution

no problem to prove => direction

for <= i can prove KH is a subset of HK

only i got troubled to show HK ia subset of KH

x in HK

x=hk for some h in H ,k in K

i manipulate it many ways and always got the form x=khk for some h in H ,k in K

HELP, and sorry no latex ,i'm very buzy now ;P

 

[snipez90]

Take b in HK so that b^-1 = hk is in HK. How do you get b back?

 

[lanedance]

so [itex] k^{-1}h^{-1} \in KH[/tex]

and $hk \in HK$, what's its inverse?

 

[annoymage]

aahhhh i see,

b in HK

b=hk for some h in H k in K

b^{-1} also is in HK

imply
$b^{-1}=(hk)^{-1}=k^{-1}h^{-1} \in KH$

right ??

 

[annoymage]

wait wrong,

for any x in HK, x^-1 in HK so x^-1=hk for some h and k

then $x=(x^{-1})^{-1}=(hk)^{-1}=k^{-1}h^{-1} \in KH$

now this is correct right?

 

[lanedance]

yeah that looks good