Quick Review of Group Theory
Key definitions
Definition: A group is a set
- There is an element
such that for all . - For all
, we have . - For all
, there is such that .
We usually just write
One can weaken these axioms in various ways and obtain an equivalent definition.
For any group
- there is only one element
satisfying axiom (1). - the regrouping in axiom (2) extends to arbitrary many elements, so the product
is well defined for any set of elements . - the inverse
for required by axiom (3) is unique.
Definition: If
Definition: If
Definition: If
- for all
, , - if
, then .
Definition: If
Definition: If
Examples
- The integers, rational numbers, real numbers, and complex numbers are all groups under addition.
- The nonzero rational numbers, real numbers, and complex numbers are all groups under multiplication.
- For
, the set of integers modulo are a group under addition. - The subset of
consisting of elements that are invertible (i.e. such that the congruence has a solution) form a group under multiplication. This group is called . If is prime, then consists of the nonzero congruence classes. - The invertible
matrices where is any of form a group under matrix multiplication. These groups come with actions on given by matrix multiplication. - For any set
, the set of bijective maps form a group under composition of functions. This group is called the symmetric group on . If there is a bijection from to , then and are isomorphic. If then is usually written and is called the symmetric group on elements. Notice that comes with an action on given by . - If
is a regular in the plane, the group of rigid motions of the plane such that form a group under composition called the Dihedral group. Dummit and Foote call this group since it has elements, but others call it since it is the symmetries of an -gon. The elements of consist of and the group law is determined by the rules with exponents read modulo , , and . The group comes with an action on the vertices of by and on the sides of by .
Cyclic Groups
Definition: A group
Proposition: Let
Corollary: A cyclic group is isomorphic either to
Properties of order: Let
- If
has infinite order for some , so do all nonzero powers of . - If
and then . - If
then the order of divides . - If
has order , then has order . - If
is cyclic of order generated by , then generates if and only if .
Proposition: The subgroups of
Euclid’s Algorithm and Congruences
Theorem: Let
Theorem: Let
has solutions if and only if
where
Remark: Notice that the congruence equation problem is equivalent to finding