1. ## Subgroup coset problem

Let G be a finite group, and suppose that A and B are subgroups of G, prove that $\displaystyle |AB| = \frac { |A||B|} { |A \bigcap B |}$

Proof so far:

Now, by the second iso theorem, $\displaystyle \frac {|AN|}{|N|} = \frac {|A|}{|A \bigcap N|}$ if N is a normal subgroup of AN.

So since B is a normal subgroup of AB, this would be true, right?

This thing look too simple, so I probably did something wrong, please check..
Thank you.

So since B is a normal subgroup of AB, this would be true, right?
This is false. Here is an counterexample.
Let $\displaystyle G=S_3$ and $\displaystyle A = \left< (123) \right>$ and $\displaystyle B = \left<(12)\right>$. Then $\displaystyle AB = S_3$ while $\displaystyle B$ is not a normal subgroup.
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Let $\displaystyle S$ to be the left cosets of $\displaystyle B$. Let $\displaystyle A$ act on $\displaystyle S$ by left translation i.e. $\displaystyle a(xB) = (ax)B$. Now consider the orbit of $\displaystyle B$ under this action. Use the theorem which says the number of elements in an orbit is equal to the index of the stabilzer in the group to get this number.

Let G be a finite group, and suppose that A and B are subgroups of G, prove that $\displaystyle |AB| = \frac { |A||B|} { |A \bigcap B |}$

$\displaystyle \boxed{1}$ let $\displaystyle a \in A, \ b \in B,$ and $\displaystyle I(a,b)=\{(x,y) \in A \times B: \ ab=xy\}.$ then: $\displaystyle |I(a,b)|=|A \cap B|.$

Proof: define $\displaystyle f: I(a,b) \longrightarrow A \cap B$ by: $\displaystyle f(x,y)=a^{-1}x.$ since $\displaystyle xy=ab,$ we have $\displaystyle a^{-1}x=by^{-1} \in B.$ thus: $\displaystyle a^{-1}x \in A \cap B.$ so $\displaystyle f$ is well-defined. now if $\displaystyle f(x_1,x_2)=f(x_2,y_2),$ then $\displaystyle a^{-1}x_1=a^{-1}x_2.$

thus $\displaystyle x_1=x_2$ and because $\displaystyle x_1y_1=x_2y_2=ab,$ we'll get $\displaystyle y_1=y_2.$ so $\displaystyle f$ is one-to-one. to prove that $\displaystyle f$ is onto, choose $\displaystyle c \in A \cap B$ and put $\displaystyle x=ac, \ y=c^{-1}b.$ obviously $\displaystyle x \in A, \ y \in B$ and $\displaystyle xy=ab.$

thus $\displaystyle (x,y) \in I(a,b)$ and $\displaystyle f(x,y)=a^{-1}x=c.$ so $\displaystyle f$ is a bijection and hence $\displaystyle |I(a,b)|=|A \cap B|. \ \ \ \square$

now suppose $\displaystyle |AB|=n$ and $\displaystyle AB=\{a_jb_j : 1 \leq j \leq n \}.$ clearly $\displaystyle A \times B = \bigcup_{j=1}^n I(a_j,b_j),$ and so using the result in $\displaystyle \boxed{1}$ we have: $\displaystyle |A||B|=|A \times B|=\sum_{j=1}^n|I(a_j,b_j)|=n|A \cap B|=|AB||A \cap B|.$