Atstract Algebra Help

[John Doe]

I actually spend about 2hrs on this question.Would sombody please help me out.
Ques: Let G be a group and H a subGroup og G.Let a,b be elements of G.
prove : If a is an element of Hb, then Ha = Hb

I Know that I have to show that Ha is a subset of Hb and vice-versa but I am
unable to get there

[matt grime]

if a is in Hb, then there exists some element h in H such that hb=a,

Then Ha=Hhb=Hb.

since h permutes the elements of H.

Of course if you really want to do it the longer way:

if x in Ha, then x=ka some k in H, since a=hb for some h, we see x=ka=khb

since kh is in H (H is a group and hence closed under composition), it follows that Ha<=Hb

Similaly if x in Hb, then x=kb some k, but since a=hb => h^{-1}a=b

x=kb=kh^{-1}a which is in Ha since H is a group and closed under inverse and composition.

[psi20]

What does it mean by :

Let I be a nonempty set of integers that is closed under addition and subtraction. Then I either consists of zero alone or else contains a smallest positive element, in which case I consists of all multiples of its smallest positive element.

What's "closed under addition and subtraction"

[bloodhound]

being closed under addition and subraction means

for any two a,b in I

a+b is in I and a-b is in I

[psi20]

thanks, that was a roadblock

[John Doe]

Thanks matt grime