Abstract Algebra Exam 2

left coset

aH={ah⌉h∈H} of G (H≤G)

right coset

Ha=ha⌉h∈H} if G (H≤G)

order of subgroup H of a finite group G

always to be a divisor of the order of G

equivalence relation

-reflexive, because a⁻¹a=e and e∈H

-symmetric, because a⁻¹b∈H => (a⁻¹b)⁻¹=>b⁻¹a

-transitive, because (a⁻¹b)(b⁻¹c)=a⁻¹c

reflexive, a~R a

a⁻¹a=e and e∈H

symmetric, a~R b => b~R a

a⁻¹b∈H => (a⁻¹b)⁻¹=>b⁻¹a

transitive, a ~R b and b ~R c => a ~R c

(a⁻¹b)(b⁻¹c)=a⁻¹c

Lagrange's Theorem

every coset (left or right) of subgroup H of a group G has same number of elements as H

The order of an element of a finite group

divides the order of the group

idex of H∈G (G:H)

Let H≤G then number of left cosets of H∈G is this

permutation

A is a function ∅:A→A that is both one to one and onto

symmetric group on n letters (Sn)

Let A be finite set {1,2,...,n} the group of an permutation of A, has n! number of elements

n! number of elements

Sn have ______ number of elements

Cayley's Theorem

Every group is isomorphic to a group of permutation

orbits

the equivalence classes in A determined by equivalence relation

cycle

A permutation is this if it has at most one orbit containing more than one element

length

the number of elements in its largest orbit