817 - Algebra

Groups

Topics in Group Theory

Sylow Theory

see also Sylow Theory

Let G be a finite group and p a prime. Write the order of \( G\) as \( |G| = p^em \) where \(p \not | m\). A Sylow p-subgroup of \( G\) is a subgroup \( H \leq G \)such that \( |H| = p^e\). That is, a Sylow p-subgroup of \(G \) is a subgroup whose order is the highest conceivable power of \(p \) according to Lagrange’s Theorem. We set \( \text{Syl}_p(G)\) to be the collection of all Sylow p-subgroups of \(G \) and \(n_p = | \text{Syl}_p(G)| \) to be the number of Sylow p-subgroups.

Semi-Direct Product

see also Semi-Direct Products

Isomorphism Theorems

see also Isomorphism Theorems

Rings

Definition: A ring is a set \(R\) with two binary operation \( +\) and \(\cdot\) satisfying:

1. \( (R,+)\) is an abelian group (with identity 0)
2. \( (R,\cdot) \) is a semigroup
3. \(\cdot \) is distributive over \( + \) (on both sides)

Ideals

A subring \( A \) of a ring \(R \) is an ideal in \(R \), if \(aR \) and \(Ra \) are subsets of \( A\) for every \(a \in A \), and \((A,+) \) is an subgroup of \((R,+) \).

Maximal Ideals

A maximal ideal of a ring \(R \) is proper ideal \(M \) such that the only ideals in \( R\) containing \(M \) are \( M\) and \(R \).

Prime Ideals

A prime ideal of a ring \(R \) is proper ideal \(P \) such that whenever \(xy\in P \) for \(x,y \in R \), \(x\in P \) or \(y \in P \).
An ideal \(P \) is prime if and only if \( R\backslash P\) is closed under multiplication.
Maximal implies prime but not conversely.

Named Rings

Euclidian Domains (EDs)

A Euclidean Domain (ED) is a domain \(R \) together with a function \(N: R \to \mathbb{Z}_{\geq 0} \) such that \(N(0)=0 \) and the following property holds: for any two elements \( a,b \in R \) with \(b\neq 0 \), there are elements \( q,r \in R\) such that \( a=qb+r \) and either \(r=0 \) or \(N(r)<N(b) \)