# 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
MA1111: Linear algebra I
Assessment Detail
This module will be examined in a 2 hour examination in Trinity term.