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make some change
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WHY-Op109 committed Jun 6, 2024
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Showing 1 changed file with 43 additions and 43 deletions.
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Expand Up @@ -3,49 +3,49 @@ title: I. Groups as an Algebraic Structure
layout: simple
---

$$\gdef{\def}{\pink{\operatorname{def}}}$$
$$\gdef{\dfs}{\pink{:=}}$$
$$\gdef{\ifif}{\pink{\iff}}$$
$$\gdef{\imp}{\pink{\implies}}$$
$$\gdef{\diff}{\pink{:\longleftrightarrow}}$$
$$\gdef{\limp}{\pink{\Longleftarrow}}$$
$$\gdef{\st}{\pink{\texttt{s.t.}}}$$
$$\gdef{\the}{\pink{\Rightarrow}}$$
$$\gdef{\wrs}{\pink{=:}}$$
$$\gdef{\typ}{\pink{:\in}}$$
$$\gdef{\if}{\pink{\texttt{if}}}$$
$$\gdef{\inner}#1#2{\left\langle\strut#1,#2\right\rangle}$$
$$\gdef{\hby}{\pink{- :}}$$
$$\gdef{\norm}#1{\left\Vert\strut#1\right\Vert}$$
$$\gdef{\abs}#1{\left\vert\strut#1\right\vert}$$
$$\gdef{\op}#1{\pink{\operatorname{#1}}}$$
$$\gdef{\ke}#1{\operatorname{Ker}(#1)}$$
$$\gdef{\gen}#1{\left \langle #1 \right \rangle}$$
$$\gdef{\geni}#1{\left( #1 \right)}$$
$$\gdef{\contra}{\pink{\texttt{Opq!}}}$$
$$\gdef{\im}#1{\operatorname{Im}(#1)}$$
$$\gdef{\ord}#1{\operatorname{ord}(#1)}$$
$$\gdef{\sgn}{\operatorname{sgn}}$$
$$\gdef{\Aut}#1{\operatorname{Aut}(#1)}$$
$$\gdef{\mod}{\operatorname{mod}}$$
$$\gdef{\ideal}{\triangleleft}$$
$$\gdef{\P}{\mathbb{P}}$$
$$\gdef{\iso}{\cong}$$
$$\gdef{\norsub}{\trianglelefteq}$$
$$\gdef{\card}#1{\operatorname{card}(#1)}$$
$$\gdef{\mul}{\cdot}$$
$$\gdef{\N}{\mathbb{N}}$$
$$\gdef{\Z}{\mathbb{Z}}$$
$$\gdef{\Q}{\mathbb{Q}}$$
$$\gdef{\R}{\mathbb{R}}$$
$$\gdef{\C}{\mathbb{C}}$$
$$\gdef{\colortext}#1{\pink{\textsf{{#1}}}}$$
$$\gdef{\colormath}#1{\pink{\mathsf{{#1}}}}$$
$$\gdef{\rank}{\mathrm{rank}}$$
$$\gdef{\null}{\mathrm{null}}$$
$$\gdef{\det}{\mathrm{det}}$$
$$\gdef{\gcd}{\mathrm{gcd}}$$
$$\gdef{\lcm}{\mathrm{lcm}}$$
$$\gdef\def{\pink{\operatorname{def}}}$$
$$\gdef\dfs{\pink{:=}}$$
$$\gdef\ifif{\pink{\iff}}$$
$$\gdef\imp{\pink{\implies}}$$
$$\gdef\diff{\pink{:\longleftrightarrow}}$$
$$\gdef\limp{\pink{\Longleftarrow}}$$
$$\gdef\st{\pink{\texttt{s.t.}}}$$
$$\gdef\the{\pink{\Rightarrow}}$$
$$\gdef\wrs{\pink{=:}}$$
$$\gdef\typ{\pink{:\in}}$$
$$\gdef\if{\pink{\texttt{if}}}$$
$$\gdef\inner#1#2{\left\langle\strut#1,#2\right\rangle}$$
$$\gdef\hby{\pink{- :}}$$
$$\gdef\norm#1{\left\Vert\strut#1\right\Vert}$$
$$\gdef\abs#1{\left\vert\strut#1\right\vert}$$
$$\gdef\op#1{\pink{\operatorname{#1}}}$$
$$\gdef\gen#1{\left \langle #1 \right \rangle}$$
$$\gdef\geni#1{\left( #1 \right)}$$
$$\gdef\contra{\pink{\texttt{Opq!}}}$$
$$\gdef\im#1{\operatorname{Im}(#1)}$$
$$\gdef\ke#1{\operatorname{Ker}(#1)}$$
$$\gdef\ord#1{\operatorname{ord}(#1)}$$
$$\gdef\sgn{\operatorname{sgn}}$$
$$\gdef\Aut#1{\operatorname{Aut}(#1)}$$
$$\gdef\mod{\operatorname{mod}}$$
$$\gdef\ideal{\triangleleft}$$
$$\gdef\P{\mathbb{P}}$$
$$\gdef\iso{\cong}$$
$$\gdef\norsub{\trianglelefteq}$$
$$\gdef\card#1{\operatorname{card}(#1)}$$
$$\gdef\mul{\cdot}$$
$$\gdef\N{\mathbb{N}}$$
$$\gdef\Z{\mathbb{Z}}$$
$$\gdef\Q{\mathbb{Q}}$$
$$\gdef\R{\mathbb{R}}$$
$$\gdef\C{\mathbb{C}}$$
$$\gdef\colortext#1{\pink{\textsf{{#1}}}}$$
$$\gdef\colormath#1{\pink{\mathsf{{#1}}}}$$
$$\gdef\rank{\mathrm{rank}}$$
$$\gdef\null{\mathrm{null}}$$
$$\gdef\det{\mathrm{det}}$$
$$\gdef\gcd{\mathrm{gcd}}$$
$$\gdef\lcm{\mathrm{lcm}}$$

### $$\colortext{1.1 Groups, Subgroups and Cosets}$$

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