Quick questions on Group Theory - Cosets / Normal Group

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October 16th 2009, 02:48 AM
aman_cc

Quick questions on Group Theory - Cosets / Normal Group

Hi - This is just to validate my understanding. You may just say yes/no to questions below (Q4 might need some explanation). If there are deeper concepts involved any pointers would we welcome.

G be a group. H is a sub-group.
Let a,b E G

Hx= Right Coset of H in G, with respect to any x E G
yH= Left Coset of H in G, with respect to any y E G

Q1. If Ha = bH, this does not imply aH = bH
Q2. However if H is Normal then above is true
Q3. If Ha = aH does not imply Hb = bH

Q4. If H in not-normal then can this happen Ha = bH (for just some specific a,b and not every a,b)

Reason why I am asking these - We know well that if every right coset is a left coset; H is Normal and vice-versa.

I am wondering if H is not normal, can it still happen that some right coset is equal to some left coset? If yes what else can we deduce?

Coset with respect to 'zero' of the group is trivial so we may exclude that.

Thanks
October 16th 2009, 07:35 AM
NonCommAlg

Quote:

Originally Posted by aman_cc

Hi - This is just to validate my understanding. You may just say yes/no to questions below (Q4 might need some explanation). If there are deeper concepts involved any pointers would we welcome.

G be a group. H is a sub-group.
Let a,b E G

Hx= Right Coset of H in G, with respect to any x E G
yH= Left Coset of H in G, with respect to any y E G

Q1. If Ha = bH, this does not imply aH = bH

this is always true! because if $Ha=bH,$ then $a=bh,$ for some $h \in H$ and thus $aH=bhH=bH.$

Quote:

Q3. If Ha = aH does not imply Hb = bH

for example let $G=S_3$ and $H=<(2 \ 3)>.$ let $a = (2 \ 3)$ and $b=(1 \ 2).$

Quote:

Q4. If H in not-normal then can this happen Ha = bH (for just some specific a,b and not every a,b)

same example of Q3 but this time choose $a=(1 \ 2 \ 3), \ b=(1 \ 3 \ 2).$
October 16th 2009, 08:08 AM
aman_cc

Quote:

Originally Posted by NonCommAlg

this is always true! because if $Ha=bH,$ then $a=bh,$ for some $h \in H$ and thus $aH=bhH=bH.$

for example let $G=S_3$ and $H=<(2 \ 3)>.$ let $a = (2 \ 3)$ and $b=(1 \ 2).$

same example of Q3 but this time choose $a=(1 \ 2 \ 3), \ b=(1 \ 3 \ 2).$

Thanks. I am sorry for some stupid questions (for e.g. Q1)

Let me plz state what all I have understood and you can correct it if I am wrong.

1) H is a normal sub-group IF AND ONLY IF every right coset of H, is a left coset of H as well.

2) If H is not a normal sub-group then there is atleast one right coset of H which is not a left coset of H

(I have an unrelated confusion here - if there is one right coset which is not a left coset, does this imply there is atleast one left coset which is not a right coset. Or these two condition are not related?)

3) $ab^{-1} \in H$ => Ha = Hb; but not aH = bH

Sorry if I am getting confused un-necessarily plz.
October 16th 2009, 08:39 AM
NonCommAlg

Quote:

Originally Posted by aman_cc

Thanks. I am sorry for some stupid questions (for e.g. Q1)

Let me plz state what all I have understood and you can correct it if I am wrong.

1) H is a normal sub-group IF AND ONLY IF every right coset of H, is a left coset of H as well.

2) If H is not a normal sub-group then there is atleast one right coset of H which is not a left coset of H

(I have an unrelated confusion here - if there is one right coset which is not a left coset, does this imply there is atleast one left coset which is not a right coset. Or these two condition are not related?)

3) $ab^{-1} \in H$ => Ha = Hb; but not aH = bH

Sorry if I am getting confused un-necessarily plz.

the answer to 1) and 2) is yes: H is a normal subgroup if and only if every right (left) coset of H is a left (right) coset of H.

for 3): we have: $ab^{-1} \in H \Longleftrightarrow Ha=Hb.$ we can have $ab^{-1} \in H$ but $aH \neq bH.$ for example let $G=S_3, \ H=<(2 \ 3)>, \ a=(1 \ 2 \ 3), \ b=(1 \ 3).$