Shreyas’ Notes

Abstract Algebra 1

MATH 356

fall, junior year

Def: bab \mid a ("bb divides aa") if a÷bZa \div b \in \mathbb{Z}

Division algorithm: a,bZa, b \in \mathbb{Z}, b>0b > 0. Then, there exist q,rZq, r \in \mathbb{Z} where 0r<b0 \leq r < b such that a=qb+ra = qb + r. Additionally, q,rq, r are unique.

Proposition: If nan \mid a and nbn \mid b, then Na+bN \mid a + b.

Proposition: If nan \mid a and cZc \in Z, then nacn \mid ac.

Definition: dd is a common divisor of aa and bb if dad \mid a and dbd \mid b.

Definition: Given a,bZa, b \in Z not both 0, we say dd is a greatest common divisor of aa and bb if:

Proposition: Given aa and bb not both 0, there is exactly one gcd of aa and bb.

Groups §

A group is a set GG together with a binary operation * on GG satisfying:

  1. associativity: for all x,y,zGx, y, z \in G, (xy)z=x(yz)(x * y) * z = x * (y * z)
  2. identity: there exists a (unique) eGe \in G such that ex=xe=xe * x = x * e = x for all xGx \in G
  3. inverse: for all xGx \in G, there exists a (unique) yGy \in G such that xy=yx=ex * y = y * x = e

If, further, we have the property that xy=yxx * y = y * x for all x,yGx, y \in G, GG is abelian.

GLn(R)={AMn×n(R):A is invertible}{GL}_n(\mathbb{R}) = \{A \in M_{n \times n}(\mathbb{R}) : A \textrm{ is invertible}\}