Topics Covered: Test 1
J C Beier
(Note: I make no promises or claims about the completeness of this document. Similarly,
there may be typos.)
Chapter 0: Preliminaries
• Note: It is not my intention to directly test this material. However, it may indirectly
be involved in problems.
• Mathematical induction: be able to accurately complete proofs using mathematical
induction.
• Equivalence relations: What is an equivalence relation? Be able to determine and proof
whether a given relation is an equivalence relation. Be able to find explicit descriptions
of equivalence classes for a given equivalence relation.
• Modular arithmetic: Be able to accurately compute using modular arithmetic.
• Functions: What is a function? Be able to show one-to-one and onto. Some useful
properties:
– Function composition is associative.
– The composition of one-to-one functions is one-to-one.
– The composition of onto functions is onto.
– If a function is one-to-one and onto, then there is an onto function that is its
inverse.
• Well Ordering Principle & the Division Algorithm: Be able to use these in proofs.
• Greatest common divisor, relatively prime, least common multiple: Know the definitions of these. Recall that the greatest common denominator of a and b can be
expressed as a linear combination of a and b. This is particularly useful with a and b
are relatively prime.
• Euclid’s Lemma & the Fundamental Theorem of Arithmetic: Useful in proofs.
• Not in Chapter 0, but fact about determinants we have found useful: det(AB) =
det(A)det(B)
1
Chapter 1: Introduction to Groups
• Terminology: group, Dihedral group, closure, identity, inverses, Abelian, associativity,
cyclic, symmetries, generators, Cayley table
• The Dihedral Group: Dn
– Understand this group from the geometric perspective
– Understand this group using generators and relations (What are its generators?
relations?)
– After Chapter 5: Understand this group from a permutations perspective
– How many elements does Dn have?
– Be able to compute/simplify in this group.
Chapter 2: Groups
• Terminology: binary operation, group (identity, inverses, Abelian, associativity, closure)
• Be able to define a group and prove that a set under a given operation is or is not a
group.
• Understand the implied operations that are not always written.
• Be comfortable with both multiplicative and additive notation.
• Examples of groups with different properties (Note: beyond simply giving examples, it
is often useful to have some prototypes in mind as you think through problems); Some
classics:
– Zn
– U (n)
– Dn
– Z, R, Q, C (under +)
– R∗, Q∗, C∗ (under ×)
– GL(n, C)
– SL(n, C)
– {1, −1, i, −i} ⊂ C under ×
– the set of translations under composition
– Non-example: Z under - (problem with associativity) or × (problem with inverses), { odd integers } under + (problem with closure)
– After Ch 4: < a >
2
– After Ch 5: Sn
• Elementary group properties:
– The identity is unique.
– Left and right cancellation laws hold.
– Inverses are unique.
– (x−1 )−1 = x
– (x−1 )n = (xn )−1
– Socks-Shoes property
Chapter 3: Finite Groups & Subgroups
• Terminology: order of a group, order of an element, subgroup, trivial subgroup, proper
subgroup, < a >, cyclic subgroup generated by a, center, centralizer
• Be able to find the order of a group and of an element.
• Be able to define a subgroup. Be able to show a subset is a subgroup.
– Be wary of the operation.
– Remember to check the set is non-empty.
– Two-step test for subgroups.
– One-step test for subgroups.
– The identity for a group must be the identity for any of its subgroups.
– Finite subgroup test (simply requires closure)
– Be able to find the subgroups
• What are the subgroups of Z? What are the implications of this?
• Be able to find the cyclic subgroup generated by a particular element.
• Be able to find the center of a group and the centralizer of an element.
• What is the relationship between the center, centralizer, and Abelian?
• Common subgroup examples (you should be able to show that these are subgroups):
– The center of a group
– The centralizer of an element
– The even numbers a subgroup of Z
– SL(n, C) a subgroup of GL(n, C)
– < a >; in particular < r >, < f >⊂ Dn
3
Chapter 4: Cyclic Groups
• Terminology: cyclic, generator, subgroup lattice, Euler phi function
• Note: Be careful about the differences in infinite cyclic groups and finite.
• Basic properties (a ∈ G where G is a cyclic group):
– When does ai = aj ?
– How does the order of an element relate to the order of the cyclic group it generates?
– What can we say about k if ak = e?
– What is the order of ak in G =< a >?
– There are options for replacing generators with other generators. In other words,
when do two elements in a cyclic group generate the same subgroup? When is an
element a generator of the whole group?You should know these.
∗ In particular, what are the generators of Zn ?
– How does the order of a relate to the order of G =< a >?
• Fundamental Theorem of Cyclic groups! Very important to know and be able to use
(and its corollary in the context of Zn )
• Be able to find all subgroups of a cyclic group. Be able to draw the subgroup lattice.
• Common examples of cyclic groups:
– Z, Zn
– < r >⊂ Dn
• Euler Phi function: Be able to compute this and use it in solving problems / proofs. The
Euler phi function is very helpful when counting elements of a given order. Understand
the implications of this. Some helpful facts:
– φ(pn ) = pn − pn−1 , p prime
– For n, m relatively prime, φ(nm) = φ(n)φ(m)
Chapter 5: Permutation Groups
• Terminology: permutation, symmetric group, Sn , cycle, transposition, m-cycle, disjoint
cycle, even, odd, An , the alternating group
• What is the relationship between Dn and Sn ?
• Be able to compute with permutations in either two-line notation or cycle notation.
• Basic properties:
4
– Is Sn Abelian?
– What is the order of Sn ?
– What does the identity look like?
– How do you find inverses?
– When do permutations commute?
– How do you find the order of an element?
– Find all possible orders.
– How do you express a permutation as the product of 2-cycles?
– All permutations are either even or odd. Which is the identity?
• A new group: The alternating group: An . What is its order?
5