An expert is a man who has made all the mistakes,
which can be made, in a very narrow field.
Bohr, Niels Henrik David (1885-1962)

Rough List of Topics:

SECOND WEEK

Definitions: (Know definitions+Understand+Examples)

Conjugate

Element

Subgroup

Automorphism of a group. The group of all such, Aut G.

Normal subgroup

Center of a group

Equivalence Relation on a set (RST)

Congruent modulo n

Left coset

Index of a subgroup in a group

Review Categories of groups and Specific groups from last week

Properties/Theorems (know, be able to use and/or review proofs):

Of conjugate elements:

a and bab^-1 have the same order

(bab^-1) ⁿ=baⁿb^-1

bab^-1=a iff a and b commute

conjugation by an element b of a group G is an automorphism of G

Of normal groups:

All subgroups abelian groups are normal

Kernels of homomorphisms are always normal

Centers are always normal subgroups

Of Equivalence Relations:

Equivalence classes are disjoint and partition the set

Of Modular Aritmetic:

How to add and multiply residue classes; how to use these properties.

Of Cosets of a subgroup H of G:

Are equivalence classes, so partition the group G. Have the same cardinality as H.

Left cosets are usually not right cosets. But if H is normal, its cosets are both left and right cosets.

Counting Formula: |G|=|H|[G:H]

Lagrange’s Thm: |H|/|G| (hence the order of any element must divide the order of the group)

Corollary: Any group of order p is the cyclic group of order p.

The fibres of a homomorphism are cosets of Ker f.

Corollary: Given group hom.f:GàG’, |G|=|Ker f| |Im f|

Results from homework problems

FIRST WEEK

Definitions: (Know+Understand+Examples)

Group (incl. definitions of associativity, identity, inverse)

Abelian

Subgroup

Proper subgroup

Order of a group

Order of an element

Cyclic group

Group generated by a set

Homomorphism

Kernel

Image

Isomorphism

Automorphism

Conjugation (incl. conjugate element, conjugate subgroup)

Normal subgroup

Center of a group

Categories of groups to be familiar with (know definition, properties, specific examples):

Cyclic groups

Cyclic finite

Cyclic infinite

Symmetric groups (includes Alternating groups)

Dihedral

Matrices, GL_n(R), SL_n(R)

Specific groups to be familiar with (definition, properties)

(Z, +), (R^x,x)

D₃ ≈ S₃

D₄

Quartenion

Klein four group

Properties/Theorems:

Of groups:

Cancellation Law

(a^-1) ^-1=a, e^-1=e, (ab)^-1=b^-1a^-1

Of homomorphisms:

Map identity to identity and inverses to inverses

Isomorphisms preserve the orders of elements

Kernels and Images are subgroups

Kernels are normal subgroups

Centers are normal subgroups

Conjugation by an element of a group G is an automorphism of G

Homework #1 (2.1-2.2)