is invertivble if there is a 
s.t.
Properties of Inverses
- If an element
has both a left inverse 
and a right inverse 
, i.e., if 
and 
, then 
, is inveritble, is its inverse - If is invertible, its inverse is unique
- Inverses multiply in the opposite order. If and
are invertible, so is the product 
, and 
. - An element may have a left inverse or a right inverse, though it is not invertible.
is a set with a law of composition which is
be elements of a group 
or if 
then 
. If 
or if 
, then b = 1.
is called the symmetric group and is denoted by 
.
) denoted 
of 
(closed under LOC)
(identity)
(inverses)
generate 
's and 
's using LoC.
be a subgroup of the additive identity 
. Either 
, or else it has the form:
has a least element.
and 
and 
?
be their greatest common divisor, the positive integer that generates the subgroup 
, i.e. 
. Then
divides both 
such that 
is a prime integer and divides 
and 
, where 
, then 
such that
, then 
divide 
. The cyclic group generated by 
is
be the cyclic subgroup of 
. Then 
. Then 
for some 
and 
are distinct elements of 
. Then 
for 
be a regular 
is
be two groups and 
a map of the underlying sets. Then 
is a homomorphism if
are elements of 
, 
and 
are group homomorphisms, then
is a group homomorphism.
, 
be the standard basis vectors for 
. Then the map from 
to its permutation matrix is a homomorphism defined as
is the sign homomorphism.
, a \in G. Then
, and 
. TFAE:
is called normal if 
, 
implies 
is also an isomorphism
of a set 
union of disjoint nonempty subsets
. An equivalence relation is required to be:
and 
, then 
Flashcard set info:
Author: CoboCards-User
Main topic: Mathematics
Topic: Abstract Algebra
School / Univ.: Rice University
City: Houston
Published: 12.02.2017
Tags: Ronan Mukamel
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