(February 2007) Fix an integer \(n\ge 2\), and write \(S_n\) for the permutation group on \(n\) letters. Let \(\varphi:S_n\to G\) be a homomorphism, where \(G\) is a group of odd order. (I.e., \(G\) is a finite group with an odd number of elements.) (a) Prove that every transposition (i.e., \(2\)-cycle) \(\tau\in S_n\) is in \(\ker\varphi\). That is, prove that \(\varphi(\tau)=e\). (b) Prove that \(\varphi\) is the trivial homomorphism; i.e., prove that for all \(\sigma\in S_n\), we have \(\varphi(\sigma)=e\).
Solution
Proof. (a) Let τ be a transposition, so o(τ) = 2. Since φ is a homomorphism, eG =φ(eSn) =φ(τ 2) =φ(τ)2, so o(φ(τ)) divides 2. But o(φ(τ)) also divides |G| by Lagrange’s Theorem. Since |G| is odd, this implies that o(φ(τ)) = 1. Thus, φ(τ) =eG. (b) We know that Sn is generated by transpositions. Given σ∈Sn, write σ as a product of transpo- sitions σ =τ1τ2··· τm. By part (a), φ(σ) =φ(τ1)φ(τ2)··· φ(τm) = (eG)m =eG as desired. QED
8 Rings Know the basic definitions: • Ring. • Commutative ring. • Ring with unity, or ring with 1. (I.e., R has a multiplicative identity, denoted 1 or 1 R.) • Field. IfR is a ring with unity, then any x∈R that has a multiplicative inversex−1 is called a unit. The set of all units of R is denoted R× and forms a group under the multiplication operation, with identity element 1 R. Some quick notes about a ring R and an element x∈R: • We have 0Rx =x0R = 0R. • Since (R, +) forms a group, for an integer n, we write nx to denote x added to itself n times (or subtracted, if n is negative; or 0 R if n = 0). • For a positive integer n, we write xn for x multiplied by itself n times. If R has unity, then x0 = 1R; if in addition x is a unit (i.e., if x has multiplicative inverse), then x−n = (x−1)n. W arnings • Don’t talk about just the “identity” of a ringR. Be clear about whether you mean the additive identity (which you should just call 0 or 0 R) or the multiplicative identity (which you should just call 1 or 1 R, assuming R is a ring with unity). • Never denote either identity of a ring by e. If you mean zero, say 0. If you mean one, say 1. Use e for the identity of a group, and use 0 and 1 for rings. • Don’t talk about just the “inverse” ofx∈R when working with rings. Make sure you are clear whether you mean the additive inverse−x (also known as the negative) or the multiplicative inversex−1 (if x is a unit, which it might not be). • Don’t confuse “unity” with “unit”. The word “unity” refers to the multiplicative identity 1 (if R even has a multiplicative identity), but “unit” refers to a element of R that has a multiplicative inverse. Polynomial Rings. If R is a commutative ring, the polynomial ring R[x] consists of all polynomials in x with coefficients in R. See 20 for a problem that involves the polynomial ring Z[x]. The important case of k[x], when k is a field, is treated in more detail in Section 13 of this Study Guide. 9 Ideals Ideals. A subset I⊆R is an ideal if it satisfies the following properties:
I⁄= ∅.
For all x,y∈I, we have x−y∈I
For all x∈I and r∈R, we have rx∈I and xr∈I. Properties 1 and 2 say that I is a subgroup of R under addition, and property 3 is sometimes informally called the “sticky” property.
Note the following: • Sometimes an ideal is called a “two-sided ideal”. (There are other objects, not on the comps syllabus, called left ideals and right ideals.) In practice, though, one almost always says just “ideal.” • When the ring is commutative, we haverx =xr, so the sticky property simplifies to: For all x∈I and r∈R, we have rx∈I. • An ideal I always contains the zero element of the ring. (Can you prove this?) • LetR be a ring with unity 1, and let I⊆R be an ideal. Then I contains 1 if and only if I =R. (Can you prove this?) See 17 and 24 for problems that uses this fact. • Some books define an ideal I to be a subring of R with the sticky property. This is a valid but redundant definition; don’t use it on the exam . It requires checking xy∈I for allx,y∈I and then checking the same thing for all x∈I and y∈R.