I am taking a first course in Abstract Algebra during the summer and have a question about how groups work.
Let's say I have a group $G$ and an element $h$ that is not in $G$. If I take any element $g$ from G, is $gh$ ever in G ? What about for all elements in $G$ with $h$ under the operation?
My intuition tells me that a group should contain all elements that are closed with all other elements under the operation.
Thanks!
Best Answer
What is $h$? If $h$ isn't an element of $G$ then it's unclear what the product $gh$ means.
One situation in which it does make sense is when $G$ is a subgroup of a group $H$, and $h \in H \setminus G$. Then $gh$ makes sense because $H$ is a group, and the product is with respect to the group operation of $H$.
In this case, if $G \leq H$, and $g \in G, h \in H \setminus G$ then $gh$ is never in $G$. If it was, $g^{-1}gh = h$ would be in $G$.