MAT 541 Fall 2003 Exam 2 Study Guide
The exam will cover chapters 3 and 4 and will be held on Monday, November 17.
Chapter 3: Quotient Groups and Homomorphisms
3.1: Definitions and
Examples
Cosets, properties of cosets, normal subgroups, relationships between the kernel
of a homomorphism and a normal subgroup, relationship between a normal
subgroup N and a quotient group G/N
3.2: Cosets and Lagranges Theorem
The statements and proofs of Lagranges theorem and its corollaries.
3.3: Isomorphism
Theorems
The statements of the 1st and 4th
isomorphism theorems, statement and proofs of the 2nd and 3rd
isomorphism theorems.
3.4: Composition
Series
Definition of composition series and solvable series,
definition of a solvable group
3.5: Transpositions
and the Alternating Group
Even and odd transpositions, the definition of the Alternating group for any n.
Chapter 4: Group Actions
4.1: Permutation
Representation
definition of a group action and its associated permutation
representation, the kernel of the action, the stabilizer of an element of the
group, the orbit of an element of the set A, the relationship between the index
of the stabilizer of an element, and its orbit.
4.2: Left
Multiplication Action, Cayley’s Theorem
The permutation representation 
associated with the
left multiplication action of a group G
on a subgroup H, the statement of Cayley’s Theorem, and its corollary.
4.3: Conjugation
Action, the Class Equation
Definitons of conjugate elements,
and conjugate sets, the statement of the Class Equation, conjugacy
in 
4.4: Automorphisms
Definitions of Aut(G), 
4.5: Sylow Theorems
Statement of Sylow’s Theorems and their applications.
Definitions: coset, quotient group, kernel of a homomorphism, normal subgroup, composition series, simple group, solvable group, alternating group, group action, stabilizer, orbit, permutation representation, p-subgroup, Sylow p-subgroup
Statements of Theorems: Lagrange’s Theorem, all isomorphism theorems, Cayley’s Theorem, Class Equation, Sylow’s Theorem
Proofs of Theorems:
Lagrange’s Theorem, 2nd and 3rd Isomorphism Theorems