Group isomorphisms, automorphisms and the set $\mathrm{Aut}(G)$

Depends on

• Monoid homomorphism and group homomorphism
• Injection, surjection, bijection

Used by

• A pushout along an isomorphism is isomorphic to the other factor Corollary
• Free products are unique up to a unique factor-compatible isomorphism Corollary
• Pure geometric braids and ordered configuration loops Corollary
• The degree of a divisor descends to the Picard group of a normal proper curve Corollary
• The trigonometric loops give π₁({(x,y):x²+y²=1},(1,0))≅ℤ Corollary
• A subgroup of an abelian group need not be characteristic Counterexample
• Cyclic sylow does not alone imply a normal p complement Counterexample
• The cyclic group ℤ/4 is not isomorphic to ℤ/2×ℤ/2 Counterexample
• Almost simple finite groups Definition
• An action of a group H on a group N by automorphisms Definition
• Artin automorphisms of the free group Definition
• Automorphisms acting on forcing names Definition
• Characteristic subgroups Definition
• Equivalence of group extensions with fixed kernel and fixed quotient Definition
• Peripheral-boundary-preserving automorphisms of Fₙ Definition
• Subnormal and normal series, factors, refinements, and equivalence Definition
• The central product G∘_α H of two groups along an isomorphism of central subgroups Definition
• The modular group of order p³ as a semidirect product C_p²⋊ Cₚ Definition
• A complex torus has a lattice of parabolic deck translations Example
• A path between basepoints induces an isomorphism of fundamental groups Example
• A pushout along an isomorphism recovers the other group Example
• Affine linear Frobenius groups over finite fields Example
• Bridge and unknotting numbers of the trefoil Example
• Point pushing one puncture around another Example
• Reducible positive semidefinite forms: factorwise treatment and the square alcove Example
• Scaling maps embed the multiplicative group of nonzero reals into the quasi-isometry group of ℤ Example
• The opposite-group functor is naturally isomorphic to the identity functor by inversion Example
• Two paths can induce distinct change-of-basepoint isomorphisms on S¹∨ S¹ Example
• False: an abelian group must have an abelian automorphism group False statement
• Actions changed by automorphisms of the kernel and complement give isomorphic semidirect products Lemma
• Classical Poincare products for A, B, D and I2(m) from the permutation and signed-permutation models Lemma
• Conjugating loop classes by a path is an isomorphism of fundamental groups Lemma
• Disconnected diagrams, direct products, and comparison of invariant forms Lemma
• P automizer condition implies fusion control Lemma
• Raising to the power 1+p is an automorphism of order p of a cyclic group of order p² Lemma
• The edge-group presentation is equivalent to the associated-subgroup presentation Lemma
• The identified subgroup used to form a central product is central, hence normal Lemma
• The interior-disc and closed-disc configuration spaces are homotopy equivalent Lemma
• The inverse of a bijective group homomorphism is a group homomorphism Lemma
• Transformation laws and S₃-action of the modular lambda function Lemma

…and 31 more results.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Judson, Abstract Algebra: Theory and Applications, Isomorphism Theorems (standard reference, not scraped)