\( \newcommand{\matr}[1] {\mathbf{#1}} \newcommand{\vertbar} {\rule[-1ex]{0.5pt}{2.5ex}} \newcommand{\horzbar} {\rule[.5ex]{2.5ex}{0.5pt}} \newcommand{\E} {\mathrm{E}} \)
deepdream of a sidewalk
Show Answer
\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\groupAdd}[1] { +_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \newcommand{\bm}[1] { \boldsymbol{#1} } \require{physics} \require{ams} \require{mathtools} \)
Math and science::Algebra::Aluffi

Cosets

Coset

Let \( G \) be a group and \( H \subset G \) a subgroup.

For any \( a \in G \), a right-coset of \( H \) is any set of the form:

[\[ ? \]]

For any \( a \in G, \) a left-coset of \( H \) is any set of the form:

[\[ ? \]]

Coset equivalence relation

A right-coset forms a block in a partition defined by an equivalence relation, \( \sim_{R} \).

[\[ x \sim_{R} y \quad \iff \quad ? \]]

A left-coset forms a block in a partition defined by a equivalence relation, \( \sim_{L} \).

[\[ x \sim_{L} y \quad \iff \quad ? \]]

The essence of the relations is to consider an element related to another if the inverse of one element brings the other element back to \( H \).