In physics, what is the importance of distinguishing between a matrix and a group? On the topic of Pauli matrices, I have noticed that some authors tend to use the term matrix and group interchangeably.
I am asking because I do not see see any profound difference referring to the matrix as a group or vice versa. For me, it is like calling a number an integer and an integer a number.
In other words, in what context do I categorize a certain matrix as belonging to a group and what are some of the implication of doing so?
 A: When you talk about a set of objects as a "group" you are emphasizing the properties of those objects relative to one another. For example,
$$\sigma_z \sigma_x = i\sigma_y \, .$$
That equation is true regardless of what vectors (states) those objects are acting on.
When you talk about objects as "operators" there's an implication that those objects operate on something.
In quantum mechanics that something is usually a quantum state such as a spin state.
If you go farther and call them "matrices" you are referring to the representation of the operators in a particular basis.
For example, when you write
$$\sigma_x = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)$$
you should read this as "the pauli $x$ operator is represented in the $z$ basis by this matrix"$^{[a]}$.
So, in summary, the word "group" emphasizes the action of the objects among themselves, the word "operator" refers to those objects as they act on quantum states, and the word "matrix" refers to the operators expressed in a particular basis.
Now, all of that said, many authors just use these words interchangeably willy-nilly, so you have to try to judge whether or not to pay special attention to their choice of wording.
$[a]$: For this reason I use the notation e.g. $[\sigma_x]_z$ to refer to matrix representations when I want to be really careful.
A: Comments to the question (v4):


*

*It seems the heart of OP's question is spurred by the unfortunate common practice in the physics literature to use the word Lie group $G$ when they are really only talking about the corresponding Lie algebra
$T_eG$, i.e. the tangent space at the identity element $e\in G$.

*Example: The Lie group $U(1)=\{z\in\mathbb{C}\mid |z|=1 \}\cong S^1$ is a circle, while the corresponding Lie algebra $u(1)=\mathbb{R}$ is the real line.

*Example: The Lie group $GL(n,\mathbb{F})=\{M\in{\rm Mat}_{n\times n}(\mathbb{F}) \mid \det(M)\neq 0\}$ has Lie algebra ${\rm Mat}_{n\times n}(\mathbb{F})$.
A: I apologize if this answer is too basic or covers things you already understand - based on your question I'm not sure exactly what your knowledge is, so I'll start from the ground up.
A "group" is a particular mathematical object that shows up a lot in physics. Groups tend to be very well-suited to describing the symmetries of a system, hence their frequent application in mathematical physics.
In particular, a group is a set $G$ together with a binary operation which satisfies a few conditions (to be listed after this paragraph). A binary operation is an operation that takes two elements of $G$ and spits out a third element of $G$ (for example, addition is a binary operation on the set of integers, since adding two integers together spits out a third integer). For the purposes of this post, I will use $*$ as a general binary operation -- that is, if $a$ and $b$ are two elements of $G$, then $a*b$ is the element obtained by performing the binary operation on $a$ and $b$.
In order for $G$ to be called a "group", it must satisfy the following four conditions:


*

*Closure - The binary operation remains within the group. That is, if $a, b \in G$, then $a*b \in G.$

*Associativity - If $a, b, c \in G$, then $(a * b) * c = a * (b * c)$.

*Identity - There exists an element $e \in G$ such that for any $a \in G$, $e * a = a * e = a.$

*Inverse - For any element $a \in G$, there exists a corresponding element $a^{-1}$ such that $a a^{-1} = a^{-1} a = e.$


For example, the integers are a group under the binary operation of addition. Since the sum of any two integers is an integer, they clearly satisfy axiom (1). Since addition of integers is associative, they satisfy axiom (2). Since $0$ is an integer and $z + 0 = 0 + z = z$ for any integer $z$, $0$ is an additive identity and so the integers satisfy axiom (3). And finally, for any integer $z$ we have another integer $(-z)$ such that $z + (-z) = (-z) + z = 0$, so the integers satisfy axiom (4).
There are plenty of groups in physics, and they show up frequently. Putting aside the mathematical formalism, a group is a set of objects that you can compose with one another to get new objects, such that each object can be negated. For example, in special relativity, the set of all boosts in one direction between reference frames forms a group - if you compose two boosts you get a third boost, and every boost can be negated by boosting in the opposite direction at the same speed.
Matrices very commonly can be collected to form mathematical groups. For example, the set of all 3x3 matrices with nonzero determinant is a group under matrix multiplication -- you can multiply them together associatively and since the determinant is nonzero each matrix has an inverse.
I mentioned previously that Lorentz boosts form a group - you may remember from special relativity that Lorentz boosts can be represented as 4x4 matrices. So this is another group made up of matrices.
Unless there is an error in the text or some edge case that I'm not thinking of, an author should not refer to a single matrix as a group - they may refer to a "matrix group," which means a group (like the set of Lorentz boosts) that is made up of a set of matrices.
It isn't like calling an integer a number or a number an integer. Just as any one integer is an element of the set "the integers," many matrices can be consider as elements of various matrix groups. A matrix group, again, is a collection of matrices, not a single matrix.
They may refer to matrix groups in order to stress that there is a whole collection of matrices that behave in the same way. Take, for example, the following Lorentz boost:
$\left( \begin{matrix} \gamma & - \beta \gamma & 0 & 0 \\ - \beta \gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right)$
I would be correct in saying that this is a single matrix. But if I wanted to refer to the collection of all such boosts, I would refer to the matrix group. For example, two valid sentences in special relativity would be:
"This Lorentz boost matrix can be used to calculate the coordinates of an event in an inertial reference frame that is moving with a constant velocity relative to the first reference frame."
"The group of Lorentz boosts can be used to calculate all possible coordinates of an event in all possible inertial reference frames that are moving with a constant velocity relative to the first reference frame."
