Question

Quotient Group:

Does anyone have a nice definition of the quotient in group theory?
e.g. what does G/H mean? I'm trying to have a clear understanding of it!

Thankyou!

Answer 1

http://www.youtube.com/watch?v=WwndchnEDS4
I don't know what that is but does this help?

Answer 2

Quotient Groups
Let H be a normal subgroup of G. Then it can be verified that the cosets of G relative to H form a group. This group is called the quotient group or factor group of G relative to H and is denoted G/H.

It can be verified that the set of self-conjugate elements of G forms an abelian group Z which is called the center of G. Note the center consists of the elements of G that commute with all the elements of G. Clearly the center is always a normal subgroup.

Theorem: A group G of order p 2 where p is prime is always abelian.

Proof: From a previous theorem, the number of invariant elements is a positive multiple of p, so the center has order p or p 2 . The latter case implies G is abelian, so consider the case ∣Z∣=p. Then ∣G/Z∣=p so G/Z is cyclic, thus we may decompose G into the cosets Z,Zg,...,Zg p−1 for some g∈G. The product of any two elements z 1 g λ,z 2 g μ is z 1 z 2 g λ+μ=z 2 g μz 1 g λ, thus G is abelian and ∣Z∣=p 2 in fact.

Define the commutator of two elements g,h of a group G by u=g −1 h −1 gh. We have u=1 if and only if gh=hg. In an abelian group, all commutators are equal to the identity. Consider the set of all commutators {u 1 ,...,u m} as g,h run through all the elements of G. This set is not necessarily closed under the group operation. We define the commutator group U to be the group generated by this set. If U=G we say G is a perfect group.

Theorem: The commutator group U of a group G is normal. G/U is abelian. U is contained in every normal subgroup that has an abelian quotient group.

Proof: Let x∈G. Then x −1 g −1 h −1 ghx=a −1 b −1 ab where a=x −1 gx,b=x −1 hx, thus U is normal.

Consider the commutator of two cosets Ux,Uy. We have

(Ux −1 )(Uy −1 )(Ux)(Uy)=Ux −1 y −1 xy=U
since x −1 y −1 xy∈U, hence G/U is abelian.

Lastly if R is any normal subgroup of G with an abelian quotient group, then for any x,y∈G we have Rx −1 y −1 xy=R since all commutators of G/R must be equal to the identity, thus R contains x −1 y −1 xy hence R⊃U.

Theorem: If A,B are normal subgroups of G with only the identity element in common then every element of A commutes with every element of B.

Proof: Consider u=a −1 b −1 ab=(a −1 b −1 a)b=a −1 (b −1 ab) where a∈A,b∈B. Then since A,B are normal, a −1 ba∈B and b −1 ab∈A, thus u∈A∩B={1 }, hence a,b commute.

Hope it gives you a clear understanding now!

Answer 3

@sarahusher is the information that I just gave you enough, and understandable or do you need more info?
Please let me know!

Answer 4

@sarahusher In group theory, a quotient group is a group obtained by aggregating similar elements of a larger group using an equivalence relation.
does it make sense to you?

Answer 5

what I would like to ask you is what level of math you are learning
so I can give you more details on it

Answer 6

Yes thankyou so much for your help!

Answer 7

No problems!

Answer 8

No probs :P :PPP :D

Answer 9

? akas?? uhm? :P

Answer 10

Nothing man!
It was kinda like a joke :PP
I didn't do ****
And you wrote a whole essay!! :D

Answer 11

haha, I know, I know!

Answer 12

XD