]> 2019-10-16T01:02:55+00:00 Elementary non-basic facts in group theory 14 en 2008-05-07T22:38:32Z 2454594.4434259 Elementary non-basic facts in group theory Conjugate-intersection index theorem 0 en Conjugate-intersection index theorem Cube map is endomorphism iff abelian (if order is not a multiple of 3) 0 en Cube map is endomorphism iff abelian (if order is not a multiple of 3) Inverse map is automorphism iff abelian 0 en Inverse map is automorphism iff abelian Normal of order equal to least prime divisor of group order implies central 0 en Normal of order equal to least prime divisor of group order implies central Poincare's theorem 0 en Poincare's theorem Product of conjugates is proper 0 en Product of conjugates is proper Square map is endomorphism iff abelian 0 en Square map is endomorphism iff abelian Trivial automorphism group implies trivial or order two 0 en Trivial automorphism group implies trivial or order two Union of all conjugates of subgroup of finite index is proper 0 en Union of all conjugates of subgroup of finite index is proper Union of two subgroups is not a subgroup unless they are comparable 0 en Union of two subgroups is not a subgroup unless they are comparable Every group is a union of cyclic subgroups 0 en Every group is a union of cyclic subgroups N-abelian iff abelian (if order is relatively prime to n(n-1)) 0 en N-abelian iff abelian (if order is relatively prime to n(n-1)) Cube map is surjective endomorphism implies abelian 0 en Cube map is surjective endomorphism implies abelian Cyclic iff not a union of proper subgroups 0 en Cyclic iff not a union of proper subgroups Conjugate-intersection index theorem 0 en Conjugate-intersection index theorem Cube map is endomorphism iff abelian (if order is not a multiple of 3) 0 en Cube map is endomorphism iff abelian (if order is not a multiple of 3) Inverse map is automorphism iff abelian 0 en Inverse map is automorphism iff abelian Normal of order equal to least prime divisor of group order implies central 0 en Normal of order equal to least prime divisor of group order implies central Poincare's theorem 0 en Poincare's theorem Product of conjugates is proper 0 en Product of conjugates is proper Square map is endomorphism iff abelian 0 en Square map is endomorphism iff abelian Trivial automorphism group implies trivial or order two 0 en Trivial automorphism group implies trivial or order two Union of all conjugates of subgroup of finite index is proper 0 en Union of all conjugates of subgroup of finite index is proper Union of two subgroups is not a subgroup unless they are comparable 0 en Union of two subgroups is not a subgroup unless they are comparable Every group is a union of cyclic subgroups 0 en Every group is a union of cyclic subgroups N-abelian iff abelian (if order is relatively prime to n(n-1)) 0 en N-abelian iff abelian (if order is relatively prime to n(n-1)) Cube map is surjective endomorphism implies abelian 0 en Cube map is surjective endomorphism implies abelian Cyclic iff not a union of proper subgroups 0 en Cyclic iff not a union of proper subgroups