Algebra II - Abstract Algebra

Exam revision

Equivalence Relations (3)

1. reflexive 2. symmetric 3. transitive

Binary Operation (3)

1. associative 2. commutative 3. identity

sr^k =

r^n-ks

Groups (3)

1. associative 2. identity element 3. inverses

Subgroups (2)

1. closed under operation 2. closed under taking inverses

in Z_n, |x_i| =

n/gcd{i,n}

|x_i+x_j| =

lcm{|x_i|, |x_j|}

Symmetric Groups: |S_n|=

n!

Symmetric Groups: |σ|=

lcm{lengths of cycles}

Homomorphisms. φ: G→H (1)

φ(g₁g₂)=φ(g₁)φ(g₂)

ker φ =

{g in G|φ(g)=e_H}

Im φ =

{h in H| some g in G, φ(g)=h}

Injective

one-to-one. ker φ = {e_G}

Surjective

onto. Im φ = H

Lagrange’s Theorem

If H≤G, then |H| divides |G|. |G|/|H| = |G:H|

Cauchy’s Theorem

Let G be A group and p a prime divisors of |G|. Then G contains an element of order p.

Sylow’s Theorem

Let G be a group and let pⁿ be the highest power of a prime p dividing |G|. Then G contains a subgroup of order pⁿ.