7
$\begingroup$

Excuse my simple question. I am just starting to learn about group theory.

I am trying to understand the description of cycle index for a permutation group. The Wikipedia entry references both the group order and the degree of the group. Also from Wikipedia, I read that

The degree of a group of permutations of a finite set is the number of elements in the set. The order of a group (of any type) is the number of elements (cardinality) in the group.

I have also read in this forum that a group can act on a set of any size. So what does it mean if a group, say $C_{3}$, acts on a set of a size not equal to a multiple of the order of the group, say a set of size $5$ in this case such as $S=\{1,2,3,4,5\}$. If we consider $C_{3}$ to be addition mod 3, then $4 \equiv 1 \pmod 3$ and $5 \equiv 2 \pmod 3$. So does $S$ have degree $5$ here even though $4$ and $5$ are equivalent to other values in the set? Since the cycle index is based on permutations perhaps my example does not make any sense. I am trying to think of an example that would use permutations but I am not quite sure how to express it mathematically.

Aside: Can anyone suggest a better source than Wikipedia for learning about cycle index?

$\endgroup$
4
$\begingroup$

When we're talking about the degree of a permutation group, the actual group structure doesn't matter. So in your example, the fact that $5 \equiv 2 \pmod 3$ is completely irrelevant; all that matters is that $C_3$ is (somehow) acting on $\{1, 2, 3, 4, 5\}$, and consequently will be said to have degree $5 = \left\lvert \{1, 2, 3, 4, 5\} \right\rvert$ in this case. In fact, if we're thinking about the action of any group on $\{1, 2, 3, 4, 5\}$, it will be said to have degree $5$, in that situation.

The term degree in this context is relative; it's not an intrinsic property of a group, but a property of the group action. It's similar to how a polynomial like $x^2 + 1$ would be considered irreducible -- over $\Bbb R$. But it can be factored as $(x - i)(x + i)$ over $\Bbb C$; the context is extremely relevant.

So, it's probably helpful to think of a permutation group as a whole bunch of data tied together: a group $G$, a set $\Omega$, and the action $G \times \Omega \to \Omega$ (or if you prefer, the homomorphism $\phi: G \to {\rm Sym}(\Omega)$) of $G$ on $\Omega$.


Example: The group of rotational symmetries of a cube is the symmetric group ${\rm Sym}(4)$, the full permutation group of the four diagonals of the cube. Thus we can think of ${\rm Sym}(4)$ as a permutation group in several ways, including but definitely not limited to:

  • A permutation group of degree $4$, if we think of it as acting on the diagonals of the cube.

  • A permutation group of degree $6$, if we think of it as acting on the faces of the cube.

  • A permutation group of degree $12$, if we think of it as acting on the edges of the cube, or

  • A permutation group of degree $8$, if we think of it as acting on the vertices of the cube.

Any subgroups of ${\rm Sym}(4)$, including $C_3$, also act on any of the above sets, and would have the same degrees in each situation.

$\endgroup$
  • $\begingroup$ Thank you. You example was very helpful. I can see know why the concept of degree versus order could be useful. $\endgroup$ – Veronica Aug 22 '15 at 4:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.