Abstract Algebra Chapter 6 definitions

automorphism

An isomorphism form a group G onto itself

Cayley's Theorem

Every group is isomorphic to a group of permutations

Inner Automorphism Induced by a

Let G be a group, and a be an element of G, the function φ_a = axa^-1 V a in G is an I.A.I. by a

Steps to prove group Isomorphism

Step 1: "Mapping." define the function φ for G -> G-bar Step 2: "1-1." Assume φ(a) = φ(b) and prove a = b Step 3: "Onto." For an element g-bar in G-bar find an element g in G s.t. φ(g) = g-bar. Step 4: "Operation Preserving." show: φ(ab) = φ(a)φ(b) V a,b in G

Properties of Isomorphisms Acting on Elements

Suppose φ is an isomporphism from G to G-bar 1. φ carries the identity of G to the Identity of G-bar 2. For every int n and for every group element a in G, φ(a^n) = [φ(a)]^n 3. For any element a & b in G, a & b commute iff φ(a) and φ(b) commute 4. G = <a> iff G-bar = <φ(a)> 5. |a| = |φ(a)| Va in G (Isomorphisms preserving orders) 6. For a fixed Int k and a fixed group of elements b in G, the equation x^k = b has the same number of solutions in G as does the equation x^k = φ(b) in G-bar 7. If G is finite, then G and G-bar have exactly the same number of elements of every order

Properties of Isomorphisms Acting in a Group

Suppose φ is an isomporphism from G to G-bar 1. φ^-1 is an isomorphism from G to G-bar 2. G is Albelian iff G-bar is Abelian 3. G is cyclic iff G-bar is cyclic 4. If K is a subgroup of G, then φ(K) = {φ(k) | k element K} is a subgroup of G-bar 5. If K-bar is a subgroup of G-bar then φ^-1(K-bar) = {g ele G | φ(g) ele of K} is a subgroup of G-bar 6. φ(Z(G)) = Z(G-bar)

Theorem 6.4 Aut(G) and Inn(G) are Groups

The set of automorphisms of a group and the set of inner automorphisms of a group are both groups under the operation of function composition.

Group Isomorphism

an isomporphism φ from a Group G to G-bar is a one to one mapping if G to G that preserves a group operation φ(ab) = φ(a)φ(b) V a,b in G Notation: G ≈ G_bar

Aut(Z_n) ≈ U(n)

for every + int n Aut(Z_n) is isomorphic to U(n)