Importance of group action in abstract algebra What are the important consequences of group action in abstract algebra? Why the action of a group on a set is defined?

We study algebraic objects by studying homomorphisms.  A group action is a homomorphism from a group to the group of automorphisms of some object.
For example:


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*for a finite set $X$ of size $n$, the set of bijections $X\to X$ is called the symmetric group $S_n$.  A group $G$ acting on $X$ is a homomorphism $G\to S_n$.  The theory of these homomorphisms is via orbits and stabilizers.

*for a topological space $X$, the group $\operatorname{Homeo}(X)$ is the set of self-homoeomorphisms of $X$.  A group acting on $X$ is a homomorphism $G\to \operatorname{Homeo}(X)$.

*Or, more geometrically, we could study homomorphisms to the group $\operatorname{Isom}(X)$ of isometries of a metric space $X$.  A dihedral group for a regular $n$-gon is the subgroup of $\operatorname{Isom}(\mathbb{R}^2)$ of isometries that send the $n$-gon to itself.

*Linear representations are group actions on vector spaces.  $\operatorname{GL}(V)$ is the group of invertible linear transformations $V\to V$, so a linear representation is a homomorphism $G\to\operatorname{GL}(V)$.
Subgroups and quotient groups can be thought of as particular kinds of group actions, too.
Group actions are just homomorphisms to a "real" group (versus to an "abstract" group).
