Actions and Sylow theorems · Definition

Group action

modern algebragroups

Before you start you need to know

Actions let a group move the elements of a set, and they are how groups appear as symmetries.

Definition

An action of a group \(G\) on a set \(X\) assigns to each \(g,h\in G\) and \(x\in X\) an element \(g\cdot x\in X\), so that the identity \(e\) moves nothing and the product of \(G\) acts by composition.

\[e\cdot x=x,\qquad (gh)\cdot x=g\cdot(h\cdot x)\]

The prototypical example is \(S_n\) acting on \(\{1,\dots,n\}\) by \(\sigma\cdot i=\sigma(i)\); the dihedral group acts on the vertices of a polygon, and any group acts on itself by left multiplication, \(g\cdot x=gx\), or by conjugation, \(g\cdot x=gxg^{-1}\).

From the two axioms it follows that each \(g\) induces a bijection \(x\mapsto g\cdot x\) of \(X\), with inverse \(x\mapsto g^{-1}\cdot x\). So an action is the same as a homomorphism from \(G\) to the group of permutations of \(X\), which connects this concept with \(S_n\) and with Cayley's theorem.

It is important to respect the order in \((gh)\cdot x=g\cdot(h\cdot x)\): first \(h\) acts and then \(g\). Formulas such as \(g\cdot x=xg\) fail this axiom in non-abelian groups. Next, actions are used to group the elements of \(X\) into orbits and to count.

Think about it

Is \(g\cdot x=xg^{-1}\) an action of \(G\) on itself?

Yes: \(e\cdot x=x\) and \(g\cdot(h\cdot x)=xh^{-1}g^{-1}=x(gh)^{-1}=(gh)\cdot x\).

Exercises 10

  1. The group \(S_3\) acts on \(X=\{1,2,3\}\) by \(g\cdot x=g(x)\). If \(g=(1\ 2\ 3)\), what is \(g\cdot1\)?
  2. In an action of \(G\) on \(X\), what is \(e\cdot x\) for \(x\in X\)?
  3. Which of the following formulas defines an action of \((\mathbb Z,+)\) on \(\mathbb Z\)?
  4. Let \(S_3\) act on \(\{1,2,3\}\), with \(g=(1\ 2\ 3)\) and \(h=(1\ 2)\). Compute \((gh)\cdot1\), composing from right to left.
  5. In \(S_3\) acting on \(\{1,2,3\}\), we have \(g\cdot1=2\) with \(g=(1\ 2\ 3)\). What is \(g^{-1}\cdot2\)?
  6. The group \(\mathbb Z_6\) acts on itself by \(g\cdot x=g+x\) (mod 6). Compute \(4\cdot5\).
  7. Let \(S_3\) act on itself by conjugation, \(g\cdot x=gxg^{-1}\). Compute \((1\ 2)\cdot(1\ 2\ 3)\).
  8. In an action of \(G\) on \(X\), what kind of function is \(x\mapsto g\cdot x\) from \(X\) to \(X\), for each \(g\in G\)?
  9. \(\mathbb Z_6\) acts on \(\mathbb Z_6\) by \(n\cdot x=x+2n\) (mod 6). How many distinct values does \(n\cdot0\) take as \(n\) ranges over \(\mathbb Z_6\)?
  10. For an arbitrary group \(G\), which formula always defines an action of \(G\) on itself?

References

  • David S. Dummit, Richard M. Foote, Abstract Algebra, 3rd edition, Wiley, 2004. §1.7
  • John B. Fraleigh, A First Course in Abstract Algebra, 7th edition, Pearson, 2002. §16