Sections, concepts, and problems

groups

definition of a group

commutative binary operations and abelian groups

bijections: one-to-one and onto functions

let X be any set; the set of all bijections from X to X is a group

subgroups

a subgroup is a subset of a group that is itself a group

generating sets for subgroups

order of a group or an element of a group

cosets

prerequisite: equivalence relations

(left) cosets

the order of a subgroup divides the order of the group

homomorphisms and isomorphisms

homomorphisms of groups (structure-preserving functions between groups)

isomorphisms of groups (bijective homomorphisms)

the kernel of a homomorphism

quotient groups and normal subgroups

normal subgroups (subgroups whose cosets form a group)

quotient groups (the resulting group from above)

the First Isomorphism Theorem (Theorem 5C)

A hand-waving overview of group theory

• A group is a set whose elements can be "multiplied" in a sensible way
• A subgroup is a sensible subcollection of these elements
• X is a generating set for G if you can get all of G by multiplying elements of X (and their inverses) together; think of X as a set of building blocks for G
• If H is a subgroup of G, then H partitions G into cells called cosets.
• Usually cosets can't be multiplied together in a well-defined manner. When they can, H is called a normal subgroup.
• A (group) homomorphism f: G -> H is just a sensible function for a group. The property that f(ab) = f(a)f(b) is what makes it sensible, since it "preserves multipication".
• If f: G -> H is a homomorphism, then the stuff that gets sent to the identity in H by f is called the kernel of f, ker(f). ker(f) is always a normal subgroup of G, and the 1st Isomorphism Theorem states that G/ker(f) is isomorphic to the image of f.