# Thread: elements in a general product

1. ## elements in a general product

Let $\displaystyle A,B$ be two subgroups of $\displaystyle G$ where only $\displaystyle A$ is normal.
If $\displaystyle x \notin AB$, does it mean that $\displaystyle x \notin A$ and $\displaystyle x \notin B$?
Suppose $\displaystyle x \notin AB$ and $\displaystyle y \notin A$. Does it implies $\displaystyle xy \notin A$?

2. ## Re: elements in a general product

First question

$\displaystyle A\subseteq \text{AB}$

therefore

$\displaystyle \text {if } x \notin\text {AB }\text {then } x \notin A$

Second question

consider $\displaystyle A=B=<e>$ the trivial group

3. ## Re: elements in a general product

if $\displaystyle x \notin A, x \notin B$, does it mean that $\displaystyle x \notin AB$?
A is normal subgroup of G.

4. ## Re: elements in a general product

Originally Posted by deniselim17
if $\displaystyle x \notin A, x \notin B$, does it mean that $\displaystyle x \notin AB$?
A is normal subgroup of G.
No

Definition: $\displaystyle \text{AB}=\{a b : a\in A, b\in B\}$ Correct?