Is group $\mathbb{R} \setminus \lbrace 0\rbrace$ together with multiplication operation a Lie group?
I'm just learning group theory and I would appreciate it if you could explain in greater detail or give a few pointers. Also, if I posted the question incorrectly, please, let me know. Thank you!
Since you are new to Lie groups, probably the part of the question that you find intimidating is proving that the multiplication map is smooth. If I give you a random manifold and a map on it, proving that the map is smooth straight from the definition is typically a little painful.
However, this manifold (call it $X$) has the practical advantage of already being an open submanifold of $\mathbb{R}^n$ (and in fact $n$ has the practical advantage of being $1$). So when we look at the multiplication map
$X \times X \to X$
we may check that it is smooth by extending that map to the map $$ \mathbb{R}^2 \to \mathbb{R} $$ given by $(x, y) \mapsto xy$. This map is certainly smooth, because you can just write down all the partials (more generally, and usefully, polynomials are smooth).
You need a similar argument for the inversion map (and there you will take advantage of the fact that $0 \not\in X$, a fact we did not need in checking this first step).
You will not need this yet, but this same way of thinking works for matrix groups: an invertible matrix is in particular a matrix; you can think of an $n$ by $n$ matrix as an element of $\mathbb{R}^{n^2}$, and the entries in a product of matrices under matrix multiplication is a polynomial in the entries of the original matrices.