Importance of group action in abstract algebra

Question

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

Answer 1

To continue along the lines of Ravi’s excellent answer:

Groups are a mathematical representation of symmetry. I’m not exaggerating at all when I say that I consider that sentence to be the most important sentence in group theory. Every time groups arise in mathematics, it’s because the structure you’re considering has some kind of symmetry to it that the group captures. Each element corresponds to a way in which the underlying object is symmetric, and the group as a whole represents all of the symmetries.

Given an object and a symmetry, you can think of a symmetry as a transformation of the object in a particular way that preserves the structure of the object. For example, physically rotating a square by 90 degrees. The action of a group on a set captures the algebraic structure of this transformation, for all the elements of the group. So, the action of a group on a set details precisely how the set transforms under the symmetry described by the group.

Answer 2

Actually, I think the question you should ask is why we deal with abstract groups as sets given some weird looking axioms (which we have become more comfortable with than say with group actions which were the original objects of study!).

Among the first groups that were studied were the permutation groups $S_n$ and you know that these groups are essentially the symmetry groups of a set with $n$ objects. These came up when trying to prove the unsolvability of the quintic and with invariant theory. In addition people were studying symmetries of geometric objects like polygons, polyhedra and the pioneers probably noticed that their symmetries always had the common principles of what we think of as an abstract group.

But the notion of an abstract group did not become a thing until much later in 1854. This wikipedia page does a great job of describing this history which hopefully will shed light on what led mathematicians to come up with group actions/groups: History of Group Theory.

Answer 3

They appear everywhere.

Every module is a special case of a set acted upon by an (abelian) group.

The Erlangen program is a whole system of thought about geometry where you think in terms of groups acting on sets.

A lot of Galois theory works like this, where automorphisms of field extensions work as group actions on the roots of a polynomial.

In general, a group action sort of encapsulates the state of a system when it is transformed with reversible transformations. For example, the configuration of cubes on a Rubik's cube, when acted upon by twisting actions (which constitute a group of reversible transformations.)

Answer 4

I’m definitely not advanced in my mathematical career to give an even semi-complete answer, and I’m sure others can give much better but here’s a start:

Well, why do we define anything in mathematics? Generally speaking,I’d say it’s because we notice some behavior in nature that we want to model or further our understanding of something in mathematics.

In the case of group actions, they can further enhance our understanding of groups via some theorems about groups such as the Sylow Theorems which (can) use groups actions to prove them. Furthermore, we can consider a group acting on itself to elucidate properties of the group.

As for group actions modeling something “natural” consider the dihedral group of order $2n$ having elements $\{r^0=e=1, r^1, r^2,\ldots, s, sr,\ldots,sr^n\}$. These act on the set of vertices of an n-sided regular polygon. I think this is a very nice example to first see if a group action and how it can model something very natural. Furthermore, group actions model lots of things in science, particularly physics (if you’re interested, google).

Please take what I’m saying with a grain of salt. I’m just an undergraduate student with a low GPA that loves math at a not so great university!

Answer 5

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:

• 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).

Answer 6

(Importance in group theory) The group action on any set to give you the new groups in form of stabilizer subgroups. In this way we can invent the new groups. (Importance in topology) The group action on any topological space to give you many quotient topological spaces. In this way we can classify many mathematical object using group action.