817 - Algebra

Groups

Topics in Group Theory

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.

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

Integral Domains

An Integral Domain, often just called domain, is a commutative ring \(R \), with \(1\neq 0 \) and has no zero divisors.

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 \(N(r)<N(b) \)

Principle Ideal Domains (PIDs)

A Principal Ideal Domain (PID) is a domain, \(R\) with the property that every ideal is principal, i.e., for each ideal \(I\), we have \(I = (a) \) for some \(a \in R\).

Unique Factorization Domains (UFDs)

A Unique Factorization Domain (UFD) is an integral domain such that every element \(r \in R\) that is non-zero and not a unit can be written as a finite product \(r = p_1 \cdots p_n \) of (not necessarily distinct) irreducible elements \(p_1,..., p_n\) of \(R\) in a way that is unique up to ordering and associates. That is, if \(r = q_1 · · · q_m\) also holds with each \(q_i\) irreducible, then \(m = n\) and there is a permutation \(\sigma \) such that, for all \(i\), we have \(p_i\) and \(q\sigma(i)\) are associates.