GAP

\(\DeclareMathOperator{\ann}{ann} \DeclareMathOperator{\Ass}{Ass} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\can}{can} \DeclareMathOperator{\Ch}{Ch} \DeclareMathOperator{\char}{char} \DeclareMathOperator{\cp}{CharPoly} \DeclareMathOperator{\codim}{codim} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\colim}{colim} \DeclareMathOperator{\cone}{cone} \DeclareMathOperator{\cont}{cont} \DeclareMathOperator{\diam}{diam} \DeclareMathOperator{\dim}{dim} \DeclareMathOperator{\edim}{embdim} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\eval}{eval} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\Frac}{Frac} \DeclareMathOperator{\Fun}{Fun} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\gcd}{gcd} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\height}{height} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\Inn}{Inn} \DeclareMathOperator{\ker}{ker} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\max}{max} \DeclareMathOperator{\Min}{Min} \DeclareMathOperator{\mp}{MinPoly} \DeclareMathOperator{\mSpec}{mSpec} \DeclareMathOperator{\Nul}{Nul} \DeclareMathOperator{\ns}{nullspace} \DeclareMathOperator{\opp}{opp} \DeclareMathOperator{\Orb}{Orb} \DeclareMathOperator{\Out}{Out} \DeclareMathOperator{\pdim}{pdim} \DeclareMathOperator{\Perm}{Perm} \DeclareMathOperator{\PtStab}{pStab} \DeclareMathOperator{\rad}{rad} \DeclareMathOperator{\range}{range} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\res}{res} \DeclareMathOperator{\sStab}{SetStab} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\SL}{SL} \DeclareMathOperator{\Span}{Span} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Stab}{Stab} \DeclareMathOperator{\Supp}{Supp} \DeclareMathOperator{\Syl}{Syl} \DeclareMathOperator{\Tor}{Tor} \DeclareMathOperator{\trace}{trace} \DeclareMathOperator{\trdeg}{trdeg} \newcommand{\Ob}{\mathrm{Ob}} \newcommand{\Set}{\mathbf{Set}} \newcommand{\Grp}{\mathbf{Grp}} \newcommand{\Ab}{\mathbf{Ab}} \newcommand{\Sgrp}{\mathbf{Sgrp}} \newcommand{\Ring}{\mathbf{Ring}} \newcommand{\Fld}{\mathbf{Fld}} \newcommand{\cRing}{\mathbf{cRing}} \newcommand{\Mod}[1]{#1-\mathbf{Mod}} \newcommand{\Cx}[1]{#1-\mathbf{Comp}} \newcommand{\vs}[1]{#1-\mathbf{vect}} \newcommand{\Vs}[1]{#1-\mathbf{Vect}} \newcommand{\vsp}[1]{#1-\mathbf{vect}^+} \newcommand{\Top}{\mathbf{Top}} \newcommand{\Setp}{\mathbf{Set}_*} \newcommand{\Alg}[1]{#1-\mathbf{Alg}} \newcommand{\cAlg}[1]{#1-\mathbf{cAlg}} \newcommand{\PO}{\mathbf{PO}} \newcommand{\Cont}{\mathrm{Cont}} \newcommand{\MaT}[1]{\mathbf{Mat}_{#1}} \newcommand{\Rep}[2]{\mathbf{Rep}_{#1}(#2)} \def\l{\lambda} \def\a{\alpha} \def\b{\beta} \def\d{\delta} \def\e{\varepsilon} \def\g{\gamma} \def\t{\theta} \def\s{\sigma} \def\z{\zeta} \def\vp{\varphi} \newcommand{\A}{\mathbb{A}} \newcommand{\B}{\mathbb{B}} \newcommand{\C}{\mathbb{C}} \newcommand{\D}{\mathbb{D}} 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\def\cR{\mathcal R} \def\cS{\mathcal S} \def\cT{\mathcal T} \def\cU{\mathcal U} \def\cV{\mathcal V} \def\cW{\mathcal W} \def\cX{\mathcal X} \def\cY{\mathcal Y} \def\cZ{\mathcal Z} \newcommand{\fa}{{\mathfrak a}} \newcommand{\fb}{{\mathfrak b}} \newcommand{\fc}{{\mathfrak c}} \newcommand{\fd}{{\mathfrak d}} \newcommand{\fe}{{\mathfrak e}} \newcommand{\ff}{{\mathfrak f}} \newcommand{\fg}{{\mathfrak g}} \newcommand{\fh}{{\mathfrak h}} \newcommand{\fi}{{\mathfrak i}} \newcommand{\fj}{{\mathfrak j}} \newcommand{\fk}{{\mathfrak k}} \newcommand{\fl}{{\mathfrak l}} \newcommand{\fm}{{\mathfrak m}} \newcommand{\fn}{{\mathfrak n}} \newcommand{\fo}{{\mathfrak o}} \newcommand{\fp}{{\mathfrak p}} \newcommand{\fq}{{\mathfrak q}} \newcommand{\fr}{{\mathfrak r}} \newcommand{\fs}{{\mathfrak s}} \newcommand{\ft}{{\mathfrak t}} \newcommand{\fu}{{\mathfrak u}} \newcommand{\fv}{{\mathfrak v}} \newcommand{\fw}{{\mathfrak w}} \newcommand{\fx}{{\mathfrak x}} \newcommand{\fy}{{\mathfrak y}} 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\def\kalg{{k-\mathrm{alg}}} \newcommand\GG[2]{\Gal(#1/#2)} \newcommand{\MF}[3]{\Mat_{#1\times #2}(#3)} \newcommand{\vectwo}[2]{\begin{bmatrix} #1 \\ #2 \end{bmatrix}} \newcommand{\vecthree}[3]{\begin{bmatrix} #1 \\ #2 \\ #3\end{bmatrix}} \def\ob{{\mathfrak{ob}} } \def\qed{\square} \def\sse{\subseteq} \def\ss{\subset} \def\ssne{\subsetneq} \def\sm{\setminus} \def\inv{^{-1}} \newcommand{\es}{\emptyset} \newcommand{\Zm}[1]{\Z/({#1})} \def\ov#1{\overline{#1}} \newcommand{\leg}[2]{\left(\frac{{#1}}{{#2}}\right)} \def\th{^{th}} \def\htpy{\simeq_{\mathrm{htpc}}} \def\textand{ \, \text{and} \, } \def\textor{ \, \text{or} \, } \def\textfor{ \, \text{for} \, } \def\textfa{ \, \text{for all} \, } \def\textst{ \, \text{such that} \, } \def\textin{ \, \text{in} \, } \def\fg{ \, \text{finitely generated} \, } \newcommand{\op}{\mathrm{op}} \newcommand{\xra}[1]{\xrightarrow{#1}} \newcommand{\xora}[1]{\xtwoheadrightarrow{#1}} \newcommand{\xira}[1]{\xhookrightarrow{#1}} \newcommand{\xla}[1]{\xleftarrow{#1}} \def\lra{\longrightarrow} \def\into{\hookrightarrow} \def\onto{\twoheadrightarrow} \newcommand{\vv}[1]{\mathbf{#1}} \newcommand{\lm}[2]{{#1}\,\l + {#2}\,\mu} \renewcommand{\v}{\vv{v}} \renewcommand{\u}{\vv{u}} \newcommand{\w}{\vv{w}} \newcommand{\x}{\vv{x}} \renewcommand{\k}{\vv{k}} \newcommand{\0}{\vv{0}} \newcommand{\1}{\vv{1}} \newcommand{\vecs}[2]{#1_1,#1_2,\dots,#1_{#2}} \newcommand{\us}[1][n]{\vecs{\u}{#1}} \newcommand{\vs}[1][n]{\vecs{\v}{#1}} \newcommand{\ws}[1][n]{\vecs{\w}{#1}} \newcommand{\vps}[1][n']{\vecs{\v'}{#1}} \newcommand{\ls}[1][n]{\vecs{\l}{#1}} \newcommand{\mus}[1][n]{\vecs{\mu}{#1}} \newcommand{\lps}[1][n]{\vecs{\l'}{#1}} \def\td{\tilde{\delta}} \def\oo{\overline{\omega}} \def\ctJ{\tilde{\mathcal J}} \def\tPhi{\tilde{\Phi}} \def\te{\tilde{e}} \def\M{\operatorname{M}} \newcommand{\homotopic}{\simeq} \newcommand{\homeq}{\cong} \newcommand{\iso}{\approx} \newcommand{\dual}{\vee} \DeclarePairedDelimiter{\abs}{|}{|} 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Section B.2 GAP

Subsection What is GAP?

Subsection Commands in GAP

CyclicGroup(3);
SymmetricGroup(3);
DihedralGroup(4); 
QuaternionGroup(8);
FreeGroup("a", "b", "c");

Size(G);
Elements(G);
Subgroups(G);
NormalSubgroups(G);
GeneratorsOfGroup(G);

DirectProduct(G1, G2);
QuotientGroup(G, N);

IsCyclic(G);
IsAbelian(G);
IsSolvable(G);
IsSimple(G);

IsSubgroup(G,H);
IsNormal(G,H);
IsCharacteristicSubgroup(G, N);

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