Category:Proofs of equivalence of definitions
From Groupprops
This is a category (or supercategory) listing a broad class of facts/results/theorems
This category lists pages that give proofs for different definitions of the same term being equivalent.
Pages in category "Proofs of equivalence of definitions"
The following 124 pages are in this category, out of 124 total. The count includes redirect pages that have been included in the category. Redirect pages are shown in italics.
A
E
- Equivalence of conjugacy and commutator definitions of normality
- Equivalence of conjugacy and coset definitions of normality
- Equivalence of definitions of 2-Engel Lie ring
- Equivalence of definitions of alternative ring
- Equivalence of definitions of Baer Lie category
- Equivalence of definitions of CA-group
- Equivalence of definitions of characteristic p-functor whose normalizer generates whole group with p'-core
- Equivalence of definitions of CN-group
- Equivalence of definitions of conjugacy functor whose normalizer generates whole group with p'-core
- Equivalence of definitions of conjugate subgroups
- Equivalence of definitions of coset
- Equivalence of definitions of cyclic group
- Equivalence of definitions of finite characteristically simple group
- Equivalence of definitions of finite nilpotent group
- Equivalence of definitions of finite solvable group
- Equivalence of definitions of finitely generated group
- Equivalence of definitions of Fitting-free group
- Equivalence of definitions of Frattini subgroup
- Equivalence of definitions of fully invariant direct factor
- Equivalence of definitions of generating set
- Equivalence of definitions of group
- Equivalence of definitions of group action
- Equivalence of definitions of group of Glauberman type for a prime
- Equivalence of definitions of group of prime power order
- Equivalence of definitions of group with finitely many homomorphisms to any finite group
- Equivalence of definitions of gyrogroup
- Equivalence of definitions of homomorphism of groups
- Equivalence of definitions of intermediately subnormal-to-normal subgroup
- Equivalence of definitions of inverse property loop
- Equivalence of definitions of left coset
- Equivalence of definitions of local powering-invariant subgroup
- Equivalence of definitions of locally cyclic periodic group
- Equivalence of definitions of locally cyclic torsion-free group
- Equivalence of definitions of locally nilpotent group that is torsion-free for a set of primes
- Equivalence of definitions of LUCS-Baer Lie group
- Equivalence of definitions of maximal among abelian subgroups
- Equivalence of definitions of maximal subgroup of group of prime power order
- Equivalence of definitions of nilpotency class
- Equivalence of definitions of nilpotent group
- Equivalence of definitions of nilpotent group that is divisible for a set of primes
- Equivalence of definitions of nilpotent group that is torsion-free for a set of primes
- Equivalence of definitions of nilpotent variety
- Equivalence of definitions of normal rank
- Equivalence of definitions of p-constrained group
- Equivalence of definitions of periodic nilpotent group
- Equivalence of definitions of periodic solvable group
- Equivalence of definitions of permuting subgroups
- Equivalence of definitions of profinite group
- Equivalence of definitions of reducible multiary group
- Equivalence of definitions of Schur-Baer variety
- Equivalence of definitions of solvable variety
- Equivalence of definitions of subgroup
- Equivalence of definitions of subgroup-conjugating automorphism
- Equivalence of definitions of subnormal subgroup
- Equivalence of definitions of subset-conjugacy-closed subgroup
- Equivalence of definitions of Sylow direct factor
- Equivalence of definitions of Sylow subgroup of normal subgroup
- Equivalence of definitions of transitively normal subgroup
- Equivalence of definitions of universal congruence condition
- Equivalence of definitions of variety-containing subgroup of finite group
- Equivalence of definitions of verbal subgroup
- Equivalence of definitions of weakly abnormal subgroup
- Equivalence of definitions of weakly closed conjugacy functor
- Equivalence of internal and external direct product
- Equivalence of normality and characteristicity conditions for conjugacy functor
- Equivalence of presentations of dicyclic group
- Every Sylow subgroup is cyclic implies metacyclic
- Extensible automorphism-invariant equals normal
- Extensible equals inner
F
- Finite group implies cyclic iff every subgroup is characteristic
- Finite groups are 1-isomorphic iff their directed power graphs are isomorphic
- Finitely generated and FC implies FZ
- Finitely many commutators implies finite derived subgroup
- Frattini's argument
- Full tetrahedral group is isomorphic to S4
I
L
- Lazard correspondence defines isomorphism of categories over Set between full subcategories of the category of groups and the category of Lie rings
- LCS-Lazard correspondence defines isomorphism of categories over Set between full subcategories of the category of groups and the category of Lie rings
- Left residual of 2-subnormal by normal is normal of characteristic
- Levi's characterization of 3-abelian groups
- Lie ring is abelian iff every subring is an ideal
- Locally cyclic iff subquotient of rationals
N
- Nilpotency class three is 3-local for Lie rings
- Nonempty alternative quasigroup equals alternative loop
- Normal equals potentially characteristic
- Normal equals retract-potentially characteristic
- Normal equals strongly image-potentially characteristic
- Normal iff potential endomorphism kernel
- Normal subgroup equals kernel of homomorphism
- Number of irreducible representations equals number of conjugacy classes
P
- P-core-free iff there exists a faithful irreducible representation over a field of characteristic p
- PGammaL(2,4) is isomorphic to S5
- PGL(2,3) is isomorphic to S4
- PGL(2,5) is isomorphic to S5
- Pi-dominating pi-subgroup implies pi-Hall
- Potentially normal-subhomomorph-containing equals normal
- PSL(2,3) is isomorphic to A4
- PSL(2,4) is isomorphic to A5
- PSL(2,5) is isomorphic to A5
- PSL(2,7) is isomorphic to PSL(3,2)
- PSL(2,9) is isomorphic to A6
- Pushforwardable equals inner
