The Symmetric Groups; Quotient Groups 1 the Group Sn

Home , Quaternion group

The Symmetric Groups; Quotient groups September 24 This material is (more than) covered in sections 6.2. 6.6, 5.7, 5.8, 4.4

1 The group Sn

• Cycles of length `, Transpositions.

• Disjoint cycles commute.

• Any element σ ∈ Sn is “essentially uniquely” expressible as a product of disjoint cycles. “Essentially unique” means that the ordering of the cycles that occur in this product doesn’t matter, and that whenever a number j ∈ [1, 2, . . . , n] is ﬁxed by σ we include the cycle of length one (j) in the product. The number of cycles that occur in the product expression for σ is the number of orbits of the subgroup generated by σ.

• Deﬁnition 1 A partition1 of n is a formula that expresses n as a sum of an ordered collection of nonincreasing positive integers

n = `1 + `2 + . . . `nu

with `1 ≥ `2 ≥ . . . `nu.

Here are the partitions of the ﬁrst few numbers:

1 = 1

2 = 2 = 1 + 1

3 =3=2+1=1+1=1

4 =4=3+1=2+2=2+1+1=1+1+1+1

5 =5=4+1=3+2=3+1+1=2+2+1=2+1+1+1=1+1+1+1+1.

• Deﬁnition 2 By P(n) let us mean the set of partitions of n. So, |P(1)| = 1, |P(2)| = 2 |P(3)| = 3 |P(4)| = 5 |P(5)| = 7,...

1This is sometimes called “unrestricted partition’” Cf. Hardy and Wright Introduction to Number Theory Fifth Edition, Oxford University Press Chapter XIX.

1 • If σ ∈ Sn is expressible as a product of disjoint cycles of lengths `1, `2, . . . , `ν and the cycles are arranged so that the order that they come is such that their lengths are nonincreasing, then n = `1 + `2 + . . . , `ν is a partition. We’ll call this partition the partition associated to sigma and we’ll denote it part(σ) ∈ P(n). This gives a mapping of sets:

part Sn−→P(n).

Exercise 1 Show that this mapping part : Sn → P(n) is surjective and that for any partition π ∈ P(n) the inverse image of π in Sn —i.e., the subset of elements of Sn that are expressible as products of disjoint cycles with lengths giving the partition π—consists of a conjugacy class of elements of the group Sn. The number of conjugacy classes in Sn is |P(n)|.

FYI: Here is an identity of power series in the variable X due to Euler:

∞ X 1 1 + |P(n)|Xn = . (1 − X)(1 − X2)(1 − X3) ... 1 There are also asymptotic approximations to P(n) of the form

q 1 π 2n √ e 3 4n 3 and even an “exact” analytic formula (due to Radmacher).

Reading: Just read through for your information and general curiopsity the lsit of exercises under the heading Computation in the Symmetric Group (you probably won’t have time to do that many of the exercises here but it is worth brooding about the collection of them).

Exercise 2 (splicing cycles) Let σ := (a1, a2, . . . , a`) and τ := (b1, b2, . . . , b`0 ) be cycles of length 0 ` and ` in Sn. Suppose that the intersection of the subsets

{a1, a2, . . . , a`} and {b1, b2, . . . , b`0 } in [1, 2, 3, . . . , n] consists of a single element. First (and this is dead easy): without loss of generality why can we suppose that it is the element

a` = b`0 ?

Now show that the product of σ and τ in either order is a cycle of length ` + `0 − 1. Give necessary 0 and suﬃcient conditions on the lengths `, ` for σ and τ to commute. Prove that Sn is generated by transpositions.

2 Exercise 3 If the partition associated to σ ∈ Sn has ν summands–i.e., if it is a partition of n of the form n = `1 + `2 + . . . `ν, then σ can be expressed as a product of n − ν transpositions.

Exercise 4 Artin page 233 exercises 15, 16

Exercise 5 For n = 4, 5, 6, 7 give the full list of orders of elements in the group Sn.

2 Quotient Groups

• Deﬁnition

• First Isomorphism Theorem

• Examples

Exercise 6 Read the deﬁnition of the Quaternion group (on top of page 48). Do Page 76, Exercises 1,7,11