# 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