]> 2026-07-03T01:40:18+00:00 Semi-basic facts in group theory 14 en Semi-basic facts in group theory Characteristically metacyclic and commutator-realizable implies abelian 0 en Characteristically metacyclic and commutator-realizable implies abelian Jordan-Holder theorem 0 en Jordan-Holder theorem Subgroup of index two is normal 0 en Subgroup of index two is normal Frattini subgroup is normal-monotone 0 en Frattini subgroup is normal-monotone Burnside's basis theorem 0 en Burnside's basis theorem Frattini subgroup is nilpotent in finite 0 en Frattini subgroup is nilpotent in finite Linearly primitive implies cyclic-center 0 en Linearly primitive implies cyclic-center Hall retract implies order-conjugate 0 en Hall retract implies order-conjugate 2-subnormal implies conjugate-permutable 0 en 2-subnormal implies conjugate-permutable K implies Frattini-free 0 en K implies Frattini-free Characteristically metacyclic and commutator-realizable implies abelian 0 en Characteristically metacyclic and commutator-realizable implies abelian Jordan-Holder theorem 0 en Jordan-Holder theorem Subgroup of index two is normal 0 en Subgroup of index two is normal Frattini subgroup is normal-monotone 0 en Frattini subgroup is normal-monotone Burnside's basis theorem 0 en Burnside's basis theorem Frattini subgroup is nilpotent in finite 0 en Frattini subgroup is nilpotent in finite Linearly primitive implies cyclic-center 0 en Linearly primitive implies cyclic-center Hall retract implies order-conjugate 0 en Hall retract implies order-conjugate 2-subnormal implies conjugate-permutable 0 en 2-subnormal implies conjugate-permutable K implies Frattini-free 0 en K implies Frattini-free INST 132 en INST