Can the composition law of a group be defined only when considering a representation or realisation of the Group?

Question

When we talk about, lets say, the Lorentz group, we define the action of the Lorentz transformation $\varLambda$ on \begin{alignat}{1} x^{\mu} & \in\mathbb{R}^{1,3},\\ x^{\mu} & \rightarrow x'^{\mu}=\varLambda^{\mu}{}_{\nu}x^{\nu},\\ \varLambda^{T}g\varLambda & =g \end{alignat} and we can show that the set of these transformations form a group and obey the composition law, \begin{align} (\varLambda_{1})^{\mu}{}_{\delta}(\varLambda_{2})^{\delta}{}_{\nu}=(\varLambda_{3})^{\mu}{}_{\nu} \end{align} In this case can we say that by defining the action of the transformations on $x^{\mu}$, we have inherently considered a representation of the Lorentz group?

If that is the case, is that is there a more fundamental notion of establishing the composition law among the abstract group elements without resorting to any kind of representations or realisations of the group?

Answer 1

In this case can we say that by defining the action of the transformations on $x^\mu$, we have inherently considered a representation of the Lorentz group?

No. The (restricted) Lorentz group $SO^+(1,3)$ is defined to be the set of $4\times 4$ matrices that preserve the components of the Minkowski metric (and are also proper and orthochronous), with the composition being matrix multiplication. This (along with its compatible differentiable manifold structure) makes it a matrix Lie group.

This definition is how the group is constructed. It does not inherently require any preexisting representation.

If that is the case, is that is there a more fundamental notion of establishing the composition law among the abstract group elements without resorting to any kind of representations or realisations of the group?

Every group needs to be defined in some way and then shown that the definition is indeed a group. There are many ways of constructing new groups from existing ones, such as products and quotients. It just happens that since this group is already constructed as a subgroup of $4\times 4$ invertible matrices, the matrices themselves are already a representation. This is known as the defining (or standard) representation. In physics, this is sometimes called the fundamental representation. (The fundamental representation in mathematics means something slightly different.) Note that there are non-matrix Lie groups as well.

Answer 2

The most abstract way of defining a group is to use a group presentation. This is a set of distinct identified group elements (known as generators) together with a set of relations between these generators. Every group has a presentation (but different presentations may define the same group).

For example, the cyclic group with $n$ elements is defined by taking a single generator $a$ and applying the relation $a^n=1$. This is notated as $\langle a | a^n=1 \rangle$ (sometimes the "$=1$" part in relations is taken as read and omitted). The dihedral group with $2n$ elements has two generators $a$ and $b$ and presentation $\langle a,b| a^n=1, b^2=1, (ab)^2=1 \rangle$.

To avoid ambiguity, we say that a presentation defines the largest possible group that satisfies the given relations - so the presentation of the cyclic group $C_2$ is $\langle a|a^2=1\rangle$, even though $C_2$ also satisfies $\langle a|a^{2m}=1\rangle$ for any integer $m$.

A group presentation defines the rules that govern composition of the group's elements in the most abstract way, and does not rely on any representation or realisation of the group. However, the relations can seem arbitrary, and it is often not the most "intuitive" way to define a group.