In beginning, I thought expressions are too long for me but finally, I am happy to learn something without confusion. Thanks. (videos still so long but... =) )
Hi, around 9:20, you mentioned that the composition law would not be well defined if N was not a normal subgroup. Why does using a normal subgroup ensure that the composition law is well defined? Where do we use the properties of a normal subgroup that help ensuring this?
This is mentioned in part 1: this is because when N is a normal subgroup the left and right coset are the same, so there is only one way to partition out G
@@tekaaable Is there an issue with extending the definition of quotient group to include non-normal subgroups and specify that the partitions are made using (say) the left coset? Or does that break something down the line?
@@rtg_onefourtwoeightfiveseven Normal Subgroup is defined in term of stable conjugation of the elements of subgroup. Means a•n•a-¹ will also be an element of subgroup N, where n is an element of the group and n is an element of subgroup N. And look carefully that he used this stable conjugation property of Normal Group to prove that the defined composition is well defined. Check that he used (b-¹•n•b). For Non-Normal groups this property isn't true therefore we can't have a stable conjugation.
I stopped the video and proved the same like this... (a•b) bar = Coset of a•b = Coset of (a•b)•n = Coset of n'•(a•b•n) [ because it's a Normal Group ] = Coset of n'•a•b•n = Coset of (n'•a)•(b•n) = Coset of a'•b' [ because n'•a is an element of Coset of a and b•n is an element of b ] = Coset of a' • Coset of b' = a' bar • b' bar Therefore, for all element of Coset of a and b the composition law is well defined.
thanks for these vids. I love the color.
In beginning, I thought expressions are too long for me but finally, I am happy to learn something without confusion. Thanks. (videos still so long but... =) )
really great, greetings from germany
Hi, around 9:20, you mentioned that the composition law would not be well defined if N was not a normal subgroup. Why does using a normal subgroup ensure that the composition law is well defined? Where do we use the properties of a normal subgroup that help ensuring this?
This is mentioned in part 1: this is because when N is a normal subgroup the left and right coset are the same, so there is only one way to partition out G
@@tekaaable Is there an issue with extending the definition of quotient group to include non-normal subgroups and specify that the partitions are made using (say) the left coset? Or does that break something down the line?
@@tekaaable Wait, never mind, just hit 8:30.
@@rtg_onefourtwoeightfiveseven Normal Subgroup is defined in term of stable conjugation of the elements of subgroup.
Means a•n•a-¹ will also be an element of subgroup N, where n is an element of the group and n is an element of subgroup N.
And look carefully that he used this stable conjugation property of Normal Group to prove that the defined composition is well defined. Check that he used (b-¹•n•b).
For Non-Normal groups this property isn't true therefore we can't have a stable conjugation.
I stopped the video and proved the same like this...
(a•b) bar
= Coset of a•b
= Coset of (a•b)•n
= Coset of n'•(a•b•n) [ because it's a Normal Group ]
= Coset of n'•a•b•n
= Coset of (n'•a)•(b•n)
= Coset of a'•b' [ because n'•a is an element of Coset of a and b•n is an element of b ]
= Coset of a' • Coset of b'
= a' bar • b' bar
Therefore, for all element of Coset of a and b the composition law is well defined.