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
• 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
• 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
• Automorphisms acting on forcing names Definition
• Characteristic subgroups Definition
• Equivalence of group extensions with fixed kernel and fixed quotient 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 path between basepoints induces an isomorphism of fundamental groups Example
• A pushout along an isomorphism recovers the other group 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
• 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 inverse of a bijective group homomorphism is a group homomorphism Lemma
• An automorphism fixing the centre pointwise induces a pairing-preserving automorphism of the central quotient, with kernel the inner automorphisms Proposition
• An automorphism of an extraspecial p-group acting trivially on its Frattini quotient is inner Proposition
• Quasi-isometries modulo bounded distance form a group, and a quasi-isometry induces an isomorphism of these groups Proposition
• The two canonical maps into a central product are injective homomorphisms whose images commute, generate it, and meet in the identified centre Proposition
• Aut(Cₙ)≅(ℤ/nℤ)^× Theorem
• Aut(ℤⁿ)≅ GLₙ(ℤ) for every finite rank n Theorem
• Cayley's theorem: every group G is isomorphic to a subgroup of Sym(G) Theorem
• Conjugation x↦ gxg⁻¹ is an automorphism Theorem
• Deg:π₁(ℝ/ℤ,[0])→(ℤ,+) is an isomorphism Theorem
• Every cyclic group is isomorphic to (ℤ,+) or to (ℤ/n,+) for its finite order n≥1 Theorem
• Finite characteristically simple groups are direct products of isomorphic simple groups Theorem
• First isomorphism theorem for groups: G/ker f congimf Theorem
• For k≥3, (ℤ/2ᵏℤ)^×≅ C₂× C_2ᵏ⁻², generated uniquely as (-1)^ε5ʲ Theorem
• For pairwise coprime positive moduli, the Chinese remainder bijection restricts to an isomorphism of unit groups Theorem
• For prime p, |Aut((ℤ/p)×(ℤ/p))|=(p²-1)(p²-p) Theorem
• Free groups on the same set are uniquely isomorphic compatibly with their generators Theorem

…and 5 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)