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second isomorphism theorem (Theorem)

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.



"second isomorphism theorem" is owned by djao.
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Attachments:
proof of second isomorphism theorem for groups (Proof) by yark
proof of second isomorphism theorem for rings (Proof) by smw
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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 8658 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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