# Algebra

## Conjugation Collapses In Commutative Groups

Exercise 3, Ch. 2, Sec. 3 - Algebra (Michael Artin): Let $$a,b$$ be elements of a group $$G$$, and let $$a' = bab^{-1}$$. Prove that $$a = a'$$ if and only if $$a$$ and $$b$$ commute.

Proof:

We have $$a = a' = bab^{-1}$$. Multiplying on the right by $$b$$ we get $$ab = ba(b^{-1}b) = ba$$ and so $$a$$ and $$b$$ commute. This argument is completely reversible so we are done.

## A Simple Statement About Conjugate Elements In A Group.

Exercise 2, Ch. 2, Sec. 3 - Algebra (Michael Artin): Prove that the products $$ab$$ and $$ba$$ are conjugate elements in a group $$G$$.

Proof:

Define $$C_{x}:G\rightarrow G$$ to be conjugation by the element $$x$$. Then $C_b(ab) = b(ab)b^{-1} = ba(bb^{-1}) = ba$ and $C_a(ba) = a(ba)a^{-1} = ab(aa^{-1}) = ab.$ Thus $$ab$$ and $$ba$$ are conjugate elements in $$G$$.

## An Isomorphism Between The Additive Reals And The Positive Reals With Multiplication

Exercise 1, Ch. 2, Sec. 3 - Algebra (Michael Artin): Prove that the additive group $${\mathbb{R}}^{+}$$ is isomorphic to the multiplicative group $$P$$ of positive reals.

Proof:

## Upper Diagonal Matrix Raised To The N-th Power

Exercise 7, Ch. 1, Sec. 1 - Algebra (Michael Artin): Find a formula for $${\left[\begin{array}{rrr} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right]}^n$$ and prove it by induction.

Proof: Defining $$A = \left[\begin{array}{rrr} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right]$$ and considering $$A^2, A^3$$ and a few other low powers of $$A$$ we come to the inductive hypothesis, 
{\left[\begin{array}{rrr}
1 & 1 & 1 \\
0 & 1 & 1 \\

## $$Z[x]$$ is not Euclidian but still a unique factorization domain

Exercise 5, Ch. 2, Sec. 2 - Fundamentals of Number Theory (William J. LeVeque):

If $$D$$ is a Euclidean domain, and $$a$$ and $$b$$ are relatively prime elements of $$D$$, then there are $$m,n \in D$$ such that $$ma + nb = 1$$.

• Show that 2 and $$x$$ are relatively prime elements of $$Z[x]$$.