Example of a group acting of a set

Question

$\quad$ In group theory class we considered only the group of geometric symmetries(rotation and reflection), but the definition involves that there can be any group and any set as long as the following conditions are met \begin{align} &1. \quad \forall g_1, g_2 \in G, \forall x \in X \quad g_1*(g_2*x) = (g_1*g_2)*x\\ &2. \quad \forall x\in X \quad e*x=x \end{align}

$\quad$ Can you provide some more examples with different groups and different sets? In particular, with $\mathbb{Z}_6$ group for instance.

Answer 1

In some sense the starting point of algebraic topology (an area of mathematics) is the action of the integers acting on the reals by translation, ie. $$\begin{array}{rcl} \Bbb Z \times \Bbb R &\rightarrow& \Bbb R\\ (n,r) & \mapsto & n+r \end{array}$$ The importance of this action is that topologically the quotient of this action can be identified with $\Bbb S^1$. This is just a fancy way to say, that if you identify all the reals, which are an integer apart, you may represent them by the unit interval $[0,1]$, but with 0 and 1 identified as well, so bending the interval and gluing its ends together gives a circle.

Studying the properties of this action lead to the theory of covering spaces, which in turn play an important motivating role in the development of homotopy theory. In fact many spaces can be identified as the quotient of a group acting on a simpler space. Examples include the torus $\Bbb T^2$ as quotient of the action $$\begin{array}{rcl} \Bbb Z^2 \times \Bbb R^2 &\rightarrow & \Bbb R^2\\ (m,n,r,s) &\mapsto &(m+r,n+s) \end{array}$$ the projective space $\Bbb RP^n$ as quotient of the action $$\begin{array}{rcl} \Bbb Z_2 \times \Bbb S^n & \rightarrow & \Bbb S^n\\ (k,x) & \mapsto & (-1)^k x \end{array}$$ and so on and so on.

Group operations occur naturally in other areas of mathematics as well. For example you may consider the automorphism groups of any field $K$. By this I mean those maps $K \rightarrow K$ that preserve addition, multiplication, the 1 and are furthermore bijective.

Take for example $K=\Bbb R$. Its automorphism group is trivial in that the only automorphism is the identity. Meanwhile the automorphism group of $\Bbb C$ consists of two automorphisms, the identity and conjugation. This tells us in particular, that $\Bbb R$ and $\Bbb C$ are nonisomorphic ie. different fields! Now every automorphism group canonically acts on its underlying field $K$ in that $$\begin{array}{rcl} \operatorname{Aut}(K) \times K & \rightarrow & K\\ ((\varphi:K\rightarrow K),x) & \mapsto & \varphi(x) \end{array}$$ For example we have the action on $\Bbb C$ given by $$\begin{array}{rcl} \{id, \overline\cdot\} \times \Bbb C & \rightarrow & \Bbb C\\ (id,x) & \mapsto & x\\ (\overline{\cdot}, x) & \mapsto & \overline{x} \end{array}$$ Now if you start cooking up more exotic fields, by adjoining certain solutions to equations to a field ($\Bbb C$ is obtained from $\Bbb R$ by forcefully adding a solution $i$ for the equation $x^2 = -1$ and closing things up under the field operations) we can obtain more interesting automorphism groups. Studying their actions (or rather the induced actions of certain subgroups) leads to a beautiful area of mathematics called Galois theory.

Finally (as it was requested) some group actions, which occur in nature.

• Any set of objects, say cards, can be equipped with a group action of the symmetric group, with action given by permutation.
• When playing with dice, you are exploiting the group of isometries of the dice acting on itself. The same holds for other types of dice used in roleplaying.
• If you happen to have long hair, you may want to braid it, which can be accomplished in many different ways. This is the braid group (of some number of strands) acting on your hair.