Problem 1.

Let’s start with some group theory! For this first question, let $G$ be a finite group of order $n$. We’d like to understand the size of the center of $G$, say $|Z(G)|=z$.
(a) Show that it is not possible for $z$ to fall in the range $\frac{n}{4}<z<n$.
(b) Show that these bounds are optimal. That is, give examples of a group where $z=n$, and one where $z=\frac{n}{4}$.

Problem 2.

1. Let $R$ be a commutative ring with $1 \neq 0$. Recall that in ideal $\mathfrak{m} \subseteq R$ is maximal if and only if $R / \mathfrak{m}$ is a field. We will see there is a similar characterization of primality.
(a) Prove that an ideal $\mathfrak{p} \subseteq R$ is prime if and only if the quotient ring $R / \mathfrak{p}$ is an integral domain.
(b) Prove that a maximal ideal $\mathfrak{m} \subseteq R$ is prime.
(c) What are all the prime ideals of $\mathbb{Z}$ ?
(d) Prove that the ideal $(x) \subseteq \mathbb{Z}[x]$ is prime but not maximal.

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# MX3020: GROUP THEORY (2020-2021)

### Course Overview

Group theory concerns the study of symmetry. The course begins with the group axioms, which provide an abstract setting for the study of symmetry. We proceed to study subgroups, normal subgroups, and group actions in various guises. Group homomorphisms are introduced and the related isomorphism theorems are proved. Composition series are introduced and the Jordan-Holder theorem is proved. Sylow p-subgroups are introduced and the three Sylow theorems are proved. Throughout symmetric groups are consulted as a source of examples.

### Qualification Prerequisites

• Either Programme Level 3 or Programme Level 4

None.

None.

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