Math 354
Modern Algebra Fall 2015
Final Exam Info
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The final exam is Friday, December 18 at 2pm in Annex 23A (our usual
classroom)
You can use a calculator
The exam is on everything we have covered, so sections 0-7, 8-11, 13, 14
Key Concepts from each section:
Section 0 Sets and Relations
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Basic Set Theory notation and terminology (symbols like ∈, , , ø, ⊆ )
What it means for a set to be well-defined
Subsets and Cartesian Products
Relations between sets, reflexive, symmetric, transitive, equivalence relation, partition
Functions, domain, range, one-to-one, onto
Cardinality, how to show two sets have the same cardinality
Section 1 Introduction and Examples
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Complex numbers: multiplication, Euler’s formula, finding solutions, algebra on
circles/nth roots of unity
Section 2 Binary Operations
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Binary operations: properties like closure, commutative, associative, composition is
always associative, what it means to be well-defined
Section 3 Isomorphic Binary Structures
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Binary algebraic structures (set with binary operation)
Isomorphisms between binary algebraic structures
Homomorphism property
Isomorphisms preserve structural properties
Identity element, uniqueness of identity element
Section 4 Groups
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Definition of group, examples of groups
Cancellation laws, unique solutions, unique identity, inverse formula
Group tables for finite groups
Different groups of order 4
Section 5 Subgroups
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The order of a group: |G|
Definition of subgroup, notation H ≤ G, improper subgroup (G), trivial subgroup ( {e} )
What you need to check to see if H is a subgroup: closed under binary operation,
identity and inverses
Cyclic subgroups <a>, generators
Section 6 Cyclic Groups
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What it means for the entire group G to be cyclic
Order of an element
Theorem: Every cyclic group is abelian
Theorem: A subgroup of a cyclic group is cyclic
The subgroups of Z
Theorem 6.10: structure of cyclic groups, order tells which group is isomorphic (Z or some
Zn), can visualize with roots of unity
Subgroups of finite cyclic groups
Section 7 Generating Sets and Cayley Diagrams
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Groups that have more than one generator
Cayley Digraphs
Section 8 Groups of Permutations
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Permutation of a set (one-to-one and onto mapping)
Permutation groups, Sn
Symmetries of equilateral triangle or symmetries of the square (don’t memorize symbols)
If G and G’ are groups, and φ is a map from G to G’, then the image φ[G] is a
subgroup of G’
Cayley’s Theorem: every group is isomorphic to a group of permutations
Section 9 Orbits, Cycles and the Alternating Groups
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Given a permutation σ, find the orbits of σ
Given σ, find the cycles
Theorem: every permutation of a finite set is a product of disjoint cycles
Transpositions, expressing cycles as products of transpositions
Theorem: no permutation can be expressed both as a product of an even number of
transpositions and as a product of an odd number
Alternating groups An form subgroup of Sn with order n!/2
Section 10 Cosets and Theorem of Lagrange
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Given a subgroup H of G, be able to find the left and right cosets of G (this partitions G
into cells that all have the same size as H)
Theorem of Lagrange: The order of a subgroup H of G is a divisor of the order of G
The index of a subgroup (G:H) = number of cosets of H = |G|/|H|
Section 11 Direct Products and Finitely Generated Abelian Groups
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Cartesian products of sets and doing binary operations component-wise to get
products of groups
Theorem: Zm X Zn is cyclic and isomorphic to Zmn if and only if m and n are relatively
prime
How to find orders of elements in groups that are products
Fundamental Theorem of Finitely Generated Abelian Groups: every finitely generated
abelian group is isomorphic to a direct product of cyclic groups
Be able to use this theorem to characterize all abelian groups, up to isomorphism, of a
given order
Section 13 Homomorphisms
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Homomorphism property, examples of homomorphisms
Properties of φ: image, inverse image, kernel
Theorem 13.12 about homomorphisms (identity, inverses, subgroups)
Theorem: If H = ker(φ), then aH = Ha = {x ∈ G | φ(x) = φ(a) }
Corollary: φ is one-to-one if and only if ker(φ) = {e}
Normal subgroups
Section 14 Factor Groups
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If H is a normal subgroup of G, then the cosets of H form a factor group G/H. Key
example is where H is the kernel of some φ.
Let H be a normal subgroup of G. Then γ:G →G/H given by γ(x) = xH is a
homomorphism with kernel H.
Fundamental Homomorphism Theorem (see 14.11 and Figure 14.10)
Equivalent conditions for a subgroup to be a normal subgroup