Abstract Algebra Chapter 6

Front 1

Isomorphism

Back 1

A one-to-one mapping φ from a group G onto another group that is operation preserving, i.e.,

φ(ab) = φ(a)φ(b) for all a, b in G.

Front 2

Isomorphic (pertaining to two groups)

Back 2

Having an isomorphism between two groups

Front 3

Cayley’s Theorem

Back 3

Every group is isomorphic to a set of permutations.

Front 4

Properties of isomorphisms acting on elements

Back 4

Suppose that φ is an isomorphism from a group G onto another group . Then

(1) φ carries the identity of G to the identity of G'
(2) For every integer n and for every element a of G, φ(a^n) = [φ(a)]^n .
(3) For elements a and b in G, a and b commute if and only if φ(a) and φ(b) commute.
(4) |a| = |φ(a)| for all a in G (isomorphisms preserve order).
(5) For a fixed integer k and a fixed group element b in G, x^k = b has the same number of solutions in G as does the equation x^k = φ(b) in G' .

Front 5

Properties of isomorphisms acting on groups

Back 5

Suppose that φ is an isomorphism from a group G onto another group . Then

(1) G is Abelian if and only if is Abelian.
(2) G is cyclic if and only if is cyclic.
(3) φ-1 is an isomorphism from onto G.
(4) If K ≤ G, then φ(K) ={ φ(k)/k e K} ≤ G' .

Front 6

Automorphism (of a group)

Back 6

An isomorphism of a group onto itself.

Front 7

Inner automorphism induced by a, denoted φa

Back 7

The function defined by φa(x) = axa-1 for all x in G.

Front 8

Aut(G) and Inn(G) are groups

Back 8

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

Front 9

For every positive integer n, Aut(Zn) ≈ U(n)

Back 9

must know