# Prove HK=KH iff HK is a Subgroup of G

• annoymage
In summary, to prove that HK = KH, it is necessary to show that HK is a subgroup of G. This can be done by proving that for any x in HK, x^-1 is also in HK, which shows that HK is closed under inversion. This can be further demonstrated by showing that for any b in HK, b^-1 is in HK, which can be written as b^-1 = k^-1h^-1 in KH. Therefore, 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

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

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

so $k^{-1}h^{-1} \in KH[/tex] and [itex] hk \in HK$, what's its inverse?

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 ??

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?

yeah that looks good

## 1. What is the definition of a subgroup?

A subgroup is a subset of a group that has the same algebraic structure as the larger group. This means that the subset must also satisfy the group axioms of closure, associativity, identity, and inverse elements.

## 2. How do you prove that HK=KH?

To prove that HK=KH, we must show that every element in HK is also in KH, and vice versa. This can be done by showing that HK is a subset of KH and KH is a subset of HK. This can be further proven by using the definitions of H and K, and the properties of subgroups.

## 3. What is the significance of proving that HK=KH?

Proving that HK=KH shows that the subgroup HK is commutative, meaning that the order in which the elements are multiplied does not affect the final result. This is an important property in mathematics and can simplify calculations and proofs.

## 4. How does proving HK=KH relate to the subgroup HK being a normal subgroup?

If HK=KH, then the subgroup HK is considered a normal subgroup of the larger group G. This means that for any element g in G, gHK=HgK, where gHK is the left coset of HK and HgK is the right coset of HK. This property is used in the definition of normal subgroups and can be proven using the commutative property of HK=KH.

## 5. Can HK be a subgroup of G if HK is not equal to KH?

No, if HK is not equal to KH, then HK is not a subgroup of G. This is because HK must satisfy the group axioms and if HK is not commutative, it does not satisfy the closure property. Therefore, HK cannot be considered a subgroup of G.

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