Institut for Matematik og Datalogi Odense Universitet

{May 9, 1996 \atop JFB}

DM11 – F96 – Lecture 14

Lecture, May 8

We began on some algebra, covering the first two sections of the notes and beginning on the third. (See the notes from lecture 13.)

Lecture, May 15

We will continue with algebra.

Lecture, May 22

We will finish with algebra, and probably briefly consider graph coloring.

Problems to be discussed in week 22

• Write the following elements of S_{4} and S_{5} as products of cycles and as prod-

ucts of transpositions. Determine their order. ( \begin{matrix}1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \end{matrix} ), ( \begin{matrix}1 & 2 & 3 & 4 & 5 \\ 5 & 2 & 4 & 3 & 1 \end{matrix} )

• Prove that A_{n} is a subgroup of S_{n} for all n.

• Prove that a cyclic group can have more than one generator.

• Give the multiplication table for the group Q which is generated by the two elements x and y which satisfy x^{4} = 1, x^{2} = y^{2}, yx = x^{3}y, and x^{2} ≠ 1. The order of Q is 8. Find all subgroups of Q. (Q is called the quaternion group).

• Let G be a group and let Z(G) = {x ∈ G | xg = gx ∀g ∈ G}. Z(G) is called the center of G. Show that Z(G) is a subgroup of G. Show that G is abelian if and only if Z(G) = G.

• Let G be a cyclic group of order n. Suppose m ∈ ZZ, m > 0, and m | n. Prove that G contains exactly one subgroup of order m.

• List the possible orders of the subgroups of ZZ_{35}^{∗}.

• Let F be a field and x be a symbol (and indeterminate). Define the ring of polynomials in the indeterminate x to be

F [x] = {a_{0} + a_{1}x + a_{2}x^{2} + ... + a_{n}x^{n} | a_{i} ∈ F ∀i, and n ≥ 0}

Addition and multiplication are defined as follows:

– If p(x) = a_{0}+a_{1}x+a_{2}x^{2}+...+a_{m}x^{m} and q(x) = b_{0}+b_{1}x+b_{2}x^{2}+...+b_{n}x^{n}, then p(x) + q(x) = c_{0} + c_{1}x + c_{2}x^{2} + ... + c_{k}x^{k}, where c_{i} = a_{i} + b_{i} for all i (any a_{i} or b_{i} which is not explicitly listed is zero).

– If p(x) = a_{0}+a_{1}x+a_{2}x^{2}+...+a_{m}x^{m} and q(x) = b_{0}+b_{1}x+b_{2}x^{2}+...+b_{n}x^{n}, then p(x) • q(x) = c_{0} + c_{1}x + c_{2}x^{2} + ... + c_{k}x^{k}, where

c_{i} = a_{i} • b_{0} + a_{i−1} • b_{1} + ... + a_{0} • b_{i}

for all i (any a_{i} or b_{i} which is not explicitly listed is zero).

Prove that F [x] is a ring.