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second isomorphism theorem
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(Theorem)
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Let $(G,*)$ be a group. Let $H$ be a subgroup of $G$ and let $K$ be a normal subgroup of $G$ . Then
- $HK := \{ h*k \mid h \in H,\ k \in K \}$ is a subgroup of $G$ ,
- $K$ is a normal subgroup of $HK$ ,
- $H \cap K$ is a normal subgroup of $H$ ,
- There is a natural group isomorphism $H/(H \cap K) = HK/K$ .
The same statement also holds in the category of modules over a fixed ring (where normality is neither needed nor relevant), and indeed can be formulated so as to hold in any abelian category.
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"second isomorphism theorem" is owned by djao.
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Cross-references: abelian category, normality, ring, fixed, modules, category, group isomorphism, normal subgroup, subgroup, group
There are 7 references to this entry.
This is version 5 of second isomorphism theorem, born on 2002-01-05, modified 2007-07-04.
Object id is 1334, canonical name is SecondIsomorphismTheorem.
Accessed 10214 times total.
Classification:
| AMS MSC: | 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties) | | | 13C99 (Commutative rings and algebras :: Theory of modules and ideals :: Miscellaneous) |
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Pending Errata and Addenda
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