On successful completion of this module, students will be able to:
Apply the notions: map/function,
surjective/injective/bijective/invertible map, equivalence relation,
partition.
Give the definition of: group, abelian group, subgroup, normal
subgroup, quotient group, direct product of groups, homomorphism,
isomorphism, kernel of a homomorphism, cyclic group, order of a group
element.
Apply group theory to integer arithmetic: define what the greatest
common divisor of two nonzero integers $m$ and $n$ is compute it and
express it as a linear combination of $n$ and $m$ using the extended
Euclidan algorithm; write down the
Cayley table of a cyclic group $\mathbb{Z}_n$ or of the multiplicative
group $(\mathbb{Z}_n)^\times$ for small $n$; determine the order of an
element of such a group.
Define what a group action is and be able to verify that
something
is a group action.
Apply group theory to describe symmetry: know the three types of
rotation symmetry axes of the cube (their "order" and how
many there are of each type); describe the elements of symmetry group
of the regular $n$-gon (the dihedral group $D_{2n}$) for small
values
of n and know how to multiply them.
Compute with the symmetric group: determine disjoint cycle form,
sign and order of a permutation; multiply two permutations.
Know how to show that a subset of a group is a subgroup or a
normal subgroup. State and apply Lagrange's theorem. State and prove
the first isomorphism theorem.
Module Content
Sets and maps.
Binary relations, equivalence relations, and partitions.
Semigroups, monoids, and groups.
Integer division; $Z_d$ as an additive group and a multiplicative
monoid.
Remainder modulo $n$ and integer division.
The symmetric group $S_n$.
Parity and the alternating group.
Generators for $S_n$.
Subgroups
Matrix groups: $GL_n$, $SL_n$, $O_n$, $SO_n$, $U_n$, $SU_n$.
The dihedral groups $D_n$ and symmetries of the cube.
Cosets and Lagrange's Theorem.
Additive subgroups of $Z$.
Greatest common divisor.
Normal subgroups and quotient groups.
Homomorphisms and the first isomorphism theorem for groups.
Multiplicative group $Z_n^*$, Fermat's little theorem and the Chinese
Remainder Theorem.
Group actions.
A Sylow theorem.
The classification of finite abelian groups.
Module Prerequisite
MA1111: Linear algebra I
Assessment Detail
This module will be examined in a 2 hour examination in Trinity term.