FoldUnfold Table of Contents The Ring of Real and Complex Numbers The Ring of Real Numbers The Ring of Complex Numbers The Ring of Real and Complex Numbers Recall from the Rings page that if $+$ and $*$ are binary operations on the set $R$, then $R$ is called a ring under $+$ and $*$ […]

## Units in a Ring

FoldUnfold Table of Contents Units in a Ring Example 1 Example 2 Units in a Ring Definition: Let $(R, +, \cdot)$ be a ring with multiplicative identity $1$. A Unit or (Multiplicatively) Invertible Element of $R$ is an element $a \in R$ for which there exists a $b \in R$ with $a \cdot b = […]

## Basic Theorems Regarding Rings

FoldUnfold Table of Contents Basic Theorems Regarding Rings Basic Theorems Regarding Rings Recall from the Rings page that a ring is a (nonempty) set $R$ with two (closed) binary operations $+$ and $\cdot$ such that: $+$ is associative, $R$ contains an additive identity element $0$ under $+$, $R$ contains an additive inverse element for each […]

## Subrings and Ring Extensions

FoldUnfold Table of Contents Subrings and Ring Extensions Subrings and Ring Extensions Definition: Let $(R, +, \cdot)$ be a ring and let $S \subseteq R$. Then $(S, +, \cdot)$ is said to be a Subring of $R$ is $S$ is a ring with respect to the $+$ and $\cdot$ defined on $R$ and $(R, +, […]

## Rings

FoldUnfold Table of Contents Rings Example 1 Example 2 Rings Recall from the Groups page that a group is a set $G$ with a (closed) binary operation $\cdot$ denoted $(G, \cdot)$ such that: 1. The operation $\cdot$ is associative, that is, for all $a, b, c \in G$ we have that $a \cdot (b \cdot […]

## Index of Common Groups

FoldUnfold Table of Contents Index of Common Groups Index of Common Groups Group Name Group Symbol Group Operation Description Size of Group Abelian/Non-Abelian Trivial Group $(\{ e \}, \cdot)$ – The simplest group consisting of only one element. $1$ Abelian The Additive Group of $\mathbb{C}$ ($\mathbb{C}, +)$ Addition The group of complex numbers under addition. […]

## The Left Regular Representation

FoldUnfold Table of Contents The Left Regular Representation The Left Regular Representation Definition: Let $G$ be a finite group. Let $\mathbb{C}[G]$ be the space of all linear combinations $\sum_{g \in G} c_g g$ where $c_g \in \mathbb{C}$ for each $g \in G$. Definition: Let $G$ be a finite group. The Left Regular Representation of $G$ […]

## Class Functions on a Group

FoldUnfold Table of Contents Class Functions on a Group Class Functions on a Group Definition: Let $G$ be a group. A Class Function on $G$ is a function $\varphi : G \to \mathbb{C}$ with the property that for all $g \in G$ and all $h \in G$ we have that $\varphi(g) = \varphi(hgh^{-1})$. As we […]

## Corollaries to the Orthogonality Theorem for Characters of Irreducible Group Representations

FoldUnfold Table of Contents Corollaries to the Orthogonality Theorem for Characters of Irreducible Group Representations Corollaries to the Orthogonality Theorem for Characters of Irreducible Group Representations Recall from The Orthogonality Theorem for Characters of Irreducible Group Representations page that if $G$ is a finite group and $V$, $W$ are representations of $G$ then: 1. If […]

## The Orthogonality Theorem for Characters of Irreducible Group Representations

FoldUnfold Table of Contents The Orthogonality Theorem for Characters of Irreducible Group Representations The Orthogonality Theorem for Characters of Irreducible Group Representations Definition: Let $G$ be a group and let $\varphi, \psi : G \to \mathbb{C}$. The Inner Product of $\varphi$ and $\psi$ will be defined as $\displaystyle{\langle \varphi, \psi \rangle = \frac{1}{|G|} \sum_{g \in […]

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