I am taking my first proof-based abstract algebra course. I did many exercises from the group theory section of my textbook. Yet, one question remains: Is the product of elements of a group in the group? If this is true, does the group have to be Abelian or be a group defined under multiplication? What restrictions must there be?
I find this question to be important since it is something that I have to assume in one of my proofs, but I am not sure whether it makes a safe assumption.
Thank you in advance for answering my question.
Be careful with "mutliplication."
To specify a group, you have to specify a set and a binary operation on that set. When we are familiar with the set and the operation, we call it by its usual name (so, in the group consisting of integers with operation being the sum of integers, we call the operation "addition"). But in the abstract, we call the operation that makes it a group "multiplication."
That means that if by "multiplication" you mean "whatever the group operation happens to be", then the answer is "yes, the product [the result of applying the group operation] of two group elements must be an element of the group."
But if by "multiplication" you mean "I have a group defined with some operation, and it just so happens that I know that one can also multiply elements of the group even though this multiplication is not the group operation", then the answer is that this operation has nothing to do with the group structure, and so you don't know whether the result of applying that irrelevant operation to two group elements will yield a group element.
For instance, consider the group whose elements consist of all $2\times 2$ matrices with real coefficients that have a $0$ in the $(2,2)$ entry: that is, all matrices of the form $$\left(\begin{array}{cc} a & b\\ c & 0 \end{array}\right)\qquad a,b,c\in\mathbb{R},$$ where the group operation is the usual matrix addition. (You should verify that this is a group).
Now, we happen to know that $2\times 2$ matrices can be multiplied to give $2\times 2$ matrices. This is knowledge we have that is independent of the group structure we have above. In this case, you could take two elements of the group, apply matrix multiplication to them, and get something that is not an element of our group: for example, if you take $$\left(\begin{array}{cc} 1 & 1\\ 1 & 0 \end{array}\right),$$ which is an element of our group, and matrix-multiply it by itself, you get $$\left(\begin{array}{cc} 2 & 1\\ 1 & 1 \end{array}\right)$$ which is not in our group. Why? Because matrix-multiplication is not the group operation. It has nothing to do with our group. In fact, one should not even consider matrix-multiplication when working with this group, because the only operation we are interested in for this group is the group operation (which is matrix-addition).