Group Theory — Subgroups & Lagrange

graph TD DefSub["Def Subgroup\nH subset G, H group under same op"] DefCoset["Def Coset\naH = ah : h in H"] DefIndex["Def Index\nG:H = number of left cosets"] DefOrdEl["Def Order of element\nord a = smallest n with a^n = e"] DefOrdGr["Def Order of group\nG = number of elements"] Crit1["Subgroup criterion 1\nH nonempty closed under product and inverse"] Crit2["Subgroup criterion 2\nH nonempty, a b inv in H for a b in H"] T1["T1 Intersection of subgroups\nis subgroup"] T2["T2 Cosets partition G\ndistinct cosets disjoint"] T3["T3 Coset size\naH = H for all a"] Lag["Lagrange\nH subgroup implies H divides G"] Cor1["Cor 1 G = G:H H"] Cor2["Cor 2 ord a divides G"] Cor3["Cor 3 G prime implies G cyclic"] DefSub --> Crit1 DefSub --> Crit2 Crit2 --> Crit1 DefSub --> T1 DefSub --> DefCoset DefSub --> DefIndex DefSub --> DefOrdGr DefCoset --> T2 DefCoset --> T3 T2 --> Lag T3 --> Lag DefIndex --> Lag DefOrdGr --> Lag Lag --> Cor1 DefIndex --> Cor1 Lag --> Cor2 DefOrdEl --> Cor2 Cor2 --> Cor3 Lag --> Cor3 classDef axiom fill:#ff6b6b,color:#fff,stroke:#c0392b classDef definition fill:#b197fc,color:#fff,stroke:#9775fa classDef theorem fill:#51cf66,color:#fff,stroke:#40c057 class DefSub,DefCoset,DefIndex,DefOrdEl,DefOrdGr definition class Crit1,Crit2,T1,T2,T3,Lag,Cor1,Cor2,Cor3 theorem