DEFINITIONS YOU NEED TO KNOW FOR MATH 365
A relation is reflexive
A relation is symmetric
A relation is transitive
An equivalence relation
Two integers are congruent modulo n
(use the definition in
exercise 0.36)
The set of complex numbers
The absolute value of a complex number
The conjugate of a complex number
The nth roots of unity
Q*, R*, C*, Z+,
Q+, R+
U, Un,
Zn
Mmxn(R), Mn(R), GL(n,R)
V (the Klein 4-group)
A binary operation on a set
A subset is closed under a binary operation
The induced operation on a subset
A binary operation is associative
A binary operation is commutative
A binary algebraic structure
Isomorphic binary structures (includes the definition of isomorphism)
A structural property of a binary structure
An identity element in a binary structure
The inverse of an element in a binary structure with identity
A group
An abelian group
the order of a group
a subgroup
proper, improper, trivial, and non-trivial subgroups
the cyclic subgroup generated by an element
a cyclic group
a generator of a group
the order of an element
a permutation
Sn, Dn, An
the identity
permutation
the orbit of an element under a permutation (use the def. on page 84)
the orbits of a permutation
a cycle
the length of a cycle
a transposition
disjoint cycles
even and odd permutations
left cosets, right cosets
the index of a subgroup in a group
a normal subgroup
a group homomorphism
the kernel of a homomorphism
a ring
a commutative ring
a ring with unity
a field