| 词条 | Fundamental theorem on homomorphisms |
| 释义 |
In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism. The homomorphism theorem is used to prove the isomorphism theorems. Group theoretic versionGiven two groups G and H and a group homomorphism f : G→H, let K be a normal subgroup in G and φ the natural surjective homomorphism G→G/K (where G/K is a quotient group). If K is a subset of ker(f) then there exists a unique homomorphism h:G/K→H such that f = h φ. In other words, the natural projection φ is universal among homomorphisms on G that map K to the identity element. The situation is described by the following commutative diagram: By setting K = ker(f) we immediately get the first isomorphism theorem. Other versionsSimilar theorems are valid for monoids, vector spaces, modules, and rings. See also
References
| last = Rose | first = John S. | contribution = 3.24 Fundamental theorem on homomorphisms | isbn = 0-486-68194-7 | mr = 1298629 | pages = 44–45 | publisher = Dover Publications, Inc., New York | title = A course on Group Theory [reprint of the 1978 original] | url = https://books.google.com/books?id=TWDCAgAAQBAJ&pg=PA44 | year = 1994}}.{{Fundamental theorems}} 2 : Theorems in abstract algebra|Fundamental theorems |
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