Order of a permutation group versus degree of a permutation group
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?
group-theory permutations
pjs36
VeronicaVeronica
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.
pjs36pjs36
$\begingroup$ 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
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