Module MA1214: Introduction to Group Theory

Credit weighting (ECTS)

5 credits

Semester/term taught

Hilary term 2012-13

Contact Hours

11 weeks, 3 lectures including tutorials per week

Lecturer

Prof. Dmitri Zaitsev

Learning Outcomes

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