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Either two cosets

Either two cosets are equal or disjoint

examples of Cayleys

examples of Cayleys theorem

Let H be

Let H be a subgroup of G, and let a belong to G. Then aH= H if and only if a belongs to H

Let H be

Let H be a subgroup of G, and let a belong to G. Then a belongs to aH

Definition Coset of

Definition Coset of H in G and example

Cayley's theorem and

Cayley's theorem and example

If K is

If K is a subgroup of G, then f(K) = {f(k) | k [ K} is a subgroup of G.

inverse of an

inverse of an isomorphism is an isomorphism from G onto G'

An isomorphism carries

An isomorphism carries cyclic group on cyclic group

|a| = |f(a)|

|a| = |f(a)| for all a in G (isomorphisms preserve orders)

If G is

If G is Abelian then G' is also Abelian

For any elements

For any elements a and b in G, a and b commute if and only if f(a) and f(b) commute

an isomorphism carries

an isomorphism carries an in G, to f(an)

An isomorphism carries

An isomorphism carries identity on identity

Definition Group Isomorphism

Definition Group Isomorphism and examples

If e= product

If e= product of r 2-cycles then r is even

If e= product

If e= product of r 2-cycles then r is even

group of even

group of even permutations

Definition of Alternating

Definition of Alternating group

Definition of odd

Definition of odd and even permutation.

if a permutation

if a permutation a can be expressed as product of 2-cycles

every permutation in

every permutation in Sn is product of 2-cycles

order of a

order of a permutation is lcm of length of disjoint cycles

disjoint cycles commute

disjoint cycles commute