First isomorphism theorem for groups: $G\mathrm{/}\ker f\cong \mathrm{im}f$

Depends on

• A homomorphism that kills a normal subgroup factors uniquely through the quotient group
• The image of a group homomorphism is a subgroup and its kernel is a normal subgroup
• A group homomorphism is injective if and only if its kernel is trivial
• Two elements have the same image under a homomorphism if and only if they lie in the same coset of its kernel
• Group isomorphisms, automorphisms and the set $\operatorname{Aut}(G)$

Used by

• G/Z(G)congInn(G) Corollary
• The centre of an extraspecial p-group has no complement Corollary
• For n≥2, reduction ℤ→ℤ/n has kernel nℤ and realises ℤ/n by the first isomorphism theorem Example
• The two-circle wedge has both regular and nonregular connected three-sheeted coverings Example
• An extension of nilpotent groups is nilpotent False statement
• A subgroup of the central quotient and its orthogonal complement have orders multiplying to the order of the quotient Lemma
• Every finite abelian group is a quotient of (ℤ/n)ᵏ for some n and k Lemma
• If p<q are primes and |G|=pq, then G has a normal subgroup of order q Lemma
• In a group extension the kernel is normal and the quotient recovers the base Lemma
• Integer abelian structure and rank by finite reduction Lemma
• Deck(E/B)≅ N_G(H)/H for a connected covering Theorem
• Every extraspecial p-group is an internal central product of nonabelian subgroups of order p³ Theorem
• Every finite abelian group is the Galois group of some finite Galois extension of ℚ Theorem
• Every group admits a presentation Theorem
• Every group of order 105 has normal Sylow 5- and 7-subgroups and is not simple Theorem
• Every group of order 30 has normal Sylow 3- and 5-subgroups and is not simple Theorem
• First isomorphism theorem for rings: R/ker f congimf Theorem
• If [G:H]=n<∞, then Core_G(H) is normal in G, [G:Core_G(H)]∣ n!, and only finitely many subgroups contain H Theorem
• Internal central products are the images of external ones Theorem
• Normal subgroups, conjugate fields, and quotient groups in the Galois correspondence Theorem
• Second isomorphism theorem for groups: H/(H∩ N)≅ HN/N Theorem
• Splitting lemma for groups: a section, a complement, and a semidirect-product decomposition are equivalent Theorem
• The kernel of the Jacobi map and the subgroup of unit squares Theorem
• The Reidemeister-Schreier presentation theorem Theorem
• Third isomorphism theorem for groups: (G/K)/(N/K)≅ G/N Theorem

Dependency tree · two levels

15 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)