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Friday, November 23, 2012 CC-BY-NC
Permutation groups and cycles

Maintainer: admin

In this lecture, we talked about permutation groups, cycles, and the order of a group. Nicolas Simard temporarily filled, as Henri Darmon was away at a conference.

1Permutation groups

A permutation of a set T is a bijection TT. If T is finite - for example, N - then the set of all permutations is denoted ΣT or Sn (we'll use the latter notation) where n=|T|.

Of course, the actual elements in T don't matter. We will refer to them as 1,2,3,n in the rest of the section, but they could really be anything and none of the properties would change.

1.1Proving that it's a group

To prove that Sn is a group, we need to show that it satisfies the following properties:

(0) If f,gSn (so they are bijections between 2 sets of n elements) then fg and gf are also bijective. Well, this is sort of a necessary condition when defining composition, so
(1) Assocativity: (Fg)h(t)=(fg)(h(t))=f(g(h(t)))=f(gh(t))=f(gh)(t) for all t.
(2) Identity element: 1R:TT is 1T(t)=t. Then 1Tf=f1T=f.
(3) f1 is the inverse of f (it can always be defined because f is bijective). Then ff1=f1f=1T.

1.2Number of elements in a permutation group

The number of elements in Sn is n!. Reason: for f(1), there are n choices. For f(2), since we need to preserve injectivity, there are n1 choices. Etc.

1.3Examples and notation

If n=1, Sn={1T} (so just the identity function).

If n=2, there are 2 functions, which we can write as follows:

1T=(1212)σ=(1221)

(The top row is the domain, the bottom row is the range. σ is just an arbitrary name for the other function in this set. Note that these are not actually matrices - this is just the notation.)

If n=3, there are 3!=6 functions, which we can enumerate as follows:

{(123123),(123132),(123213),(123231),(123312),(123321)}

1.4Composition

Composing two functions in a permutation group is easy. Let's look at an example in S5.

f=(1234523451)g=(1234554321)

then, their composition fg is found by mapping each element to its image under g, then under f:

fg=(1234515432)

Incidentally, composition is only commutative in S1 and S2; for all other n, it is not.

2Cycles

The notation for a cycle is σ=(1,2,3). This tells us that 12, 23, 31 in a permutation group. Any other element in Sn but not mentioned in the cycle simply stays the same. Note that this is equivalent to (2,3,1) which is equivalent to (3,1,2) (so cycles are invariant under rotation or something).

Another example of a cycle: (4,1) which means that 41, 14, 22 and 33. Incidentally, a cycle of two elements (like this one) is called a transposition.

2.1Cycle decomposition

We can decompose any permutation group into cycles. Some examples:

(1234523154)=(1,2,3)(4,5)

(123456326145)=(1,3,6,5,4)(2)=(1,3,6,5,4)

In the latter example, the (2) at the end is not necessary to write so we can just omit it.

3Order

The order of a group G is the number of elements in G. We can write this as |G| or #G. So |Sn|=n!, |Z/nZ|=n. We can think of it as the cardinality of the set under G, basically, but the notion of an order is separate.

The order of an element in G is the size of its cycle. We can write it as |<g>| or |#<g>|. If G=<g> for some g, then G is cyclic (i.e., it's generated by one cycle? Not completely sure about the notation here).

Example: the group (Z,+) is cyclic, generated by one.