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2-subnormal implies join-transitively subnormal, ACIC is characteristic subgroup-closed, Artin's induction theorem, Ascending chain condition on subnormal subgroups implies subnormal join property, Automorph-conjugacy is transitive, Automorphism sends more than three-fourths of elements to inverses implies abelian, Cartan-Brauer-Hua theorem, Central implies normal satisfying the subgroup-to-quotient powering-invariance implication, Centralizer product theorem, Characteristic of normal implies normal, Characteristicity is transitive, Classification of groups of prime-cube order, Classification of groups of prime-square order, Commensurator of subgroup is subgroup, Commuting of non-identity elements defines an equivalence relation between prime divisors of the order of a finite CN-group, Complemented normal implies quotient-powering-invariant, Congruence condition on index of subgroup containing Sylow-normalizer, Conjugacy class of elements with semisimple generalized Jordan block does not split in special linear group over a finite field, Conjugacy class of prime power size implies not simple, Corollary of centralizer product theorem for rank at least three, Cube map is surjective endomorphism implies abelian, Degree of irreducible representation divides index of abelian normal subgroup, Degree of irreducible representation divides order of group, Derived subgroup satisfies ascending chain condition on subnormal subgroups implies subnormal join property, Endomorphism kernel implies quotient-powering-invariant, Endomorphism sends more than three-fourths of elements to squares implies abelian, Equivalence of definitions of gyrogroup, Every finite division ring is a field, Finite and any two maximal subgroups intersect trivially implies not simple non-abelian, Finite group having at least two conjugacy classes of involutions has order less than the cube of the maximum of orders of centralizers of involutions, Finite non-abelian and every proper subgroup is abelian implies metabelian, Finite non-abelian and every proper subgroup is abelian implies not simple, Finite non-nilpotent and every proper subgroup is nilpotent implies not simple, Finite supersolvable implies subgroups of all orders dividing the group order, Finitely generated and residually finite implies Hopfian, Fixed-point-free involution on finite group is inverse map, Frattini-in-center odd-order p-group implies (mp plus 1)-power map is automorphism, Frattini-in-center odd-order p-group implies p-power map is endomorphism, Hypoabelian not implies imperfect, Infinite group with cofinite topology is not a topological group, Isoclinic groups have same nilpotency class, Isoclinic groups have same proportions of conjugacy class sizes, Linear representation is realizable over principal ideal domain iff it is realizable over field of fractions, Local finiteness is extension-closed, Monomorphism iff injective in the category of groups, Nilpotent Hall subgroups of same order are conjugate, Normal of finite index implies completely divisibility-closed, Normal of finite index implies quotient-powering-invariant, Normal subgroup contained in the hypercenter satisfies the subgroup-to-quotient powering-invariance implication, Normalizing join-closed subgroup property implies every maximal element is intermediately subnormal-to-normal