Question about group action Can someone explain what a group action is in layman terms? I know what a group is, but I can't understand what a group action, even after doing a lot of Google searches. 
 A: One of the common ways that groups appear in mathematics is as symmetry groups. 
This is very common in geometry. For example, the symmetries of a regular $n$-gon form a group isomorphic to the dihedral group of order $2n$. Also, the symmetries of a regular tetrahedron form a group isomorphic to the symmetric group on 4 elements, which has order $4!=24$. More generally the symmetries of a regular $n$-simplex in $n$-dimensional space form a group isomorphic to the symmetric group on $n+1$ elements, which has order $(n+1)!$.
This is also common in other branches of mathematics. For example, in Galois theory, the symmetries of the roots of a polynomial that are induced by automorphisms of the splitting field form the Galois group of that polynomial (which, historically, was the very first kind of group explicitly noticed by mathematicians; although one might argue that the 17 wallpaper groups were noticed earlier by medieval artisans).
From a mathematical historical perspective, groups of symmetries appeared before abstract groups. Once the concept of a group is abstracted, then of course the power of algebra kicks in, and we can make great strides in understanding group theory. Nonetheless, many of the practical applications of groups are still rooted in their origins as groups of symmetries.
So, how can one connect the original notion of symmetry groups with the abstract notion of a group? 
The answer is: with the concept of a group action. When you are given a group $G$ and a mathematical object $X$ (like a tetrahedron, or the set of roots of a polynomial, or any mathematical object whatsoever), you can ask questions like:


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*Is this group $G$ the group of symmetries of $X$?


This rather informal question can be formalized by rewording it as: 


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*Is there an isomorphism between $G$ and the group of symmetries of $X$?


As is often fruitful, it might be useful to broaden one's interest from isomorphisms to homomorphisms:


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*Is there a homomorphism from $G$ to the group of symmetries of $X$?


And now one is ready for the definition: 


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*An action of a group $G$ on a mathematical object $X$ is a homomorphism from $G$ to the group of symmetries of $X$.


If this definition does not sound familiar, go compare it to whatever definition you like, such as the definitions in other answers here. You will find that these definitions are exactly equivalent.
A: We associate each group element $g$ with an own function $f_g$. Each such function takes elements from a set $X$ and puts back something from the same set. 
Two particular things must be fulfilled for the set of functions:


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*Identity: The function corresponding to the identity element of the group must do $x\to x, \forall x\in X$

*Compatibility requires that $f_{ab} = f_a \circ f_b$. That is group operation "translates" to function composition for the group action. $f_{ab}(x) = f_a(f_b(x))$

A: You need to first understand endofunction composition and invertible endofunctions. Suppose we have a set $X$ and a family $F$ of endofunctions $f:X\to X$. Given two such functions, $f_1, f_2\in F$ the composition $f_3$ is given by $f_3(x) := f_1(f_2(x))$. We require that the composition of endofunctions in the family is also in the family. An invertible endofunction $f_1$ is such that there is an inverse endofunction $f_2$ such that $f_2(f_1(x)) = f_1(f_2(x)) = x$ for all $x\in X$. We require that the inverse of any endofunction in the family is also in the family. By the definition of a group, the family $F$ is a group because composition is associative and so on.
Now, suppose we have an abstract group $G$ and a mapping $\phi:G\to F$ that is compatible between the group operation and endofunction composition. That is, if $\;\phi(g_1)=f_1,\;\phi(g_2)=f_2\;$ and $\phi(g_1g_2)=f_3,\;$ then $f_2(f_1(x))=f_3(x)$. We say that $\phi$ is a homomorphism of groups, but more specifically, in this case, it is a group $G$ acting (on the left) of the set $X$ via $\varphi:G\times X\to X$ where $\varphi(g,x)=\phi(g)(x)$. This idea of mapping an abstract group element to an endofunction is very common in applications of group theory to science. Often, an invertible endofunction is called a permutation and that is an alternative name for it.
