Math 2.0

Group theory and generalizations Definition

Internal direct product

Builds on Direct product of groupsNormal subgroupProduct of two subgroups

Definition

Let GG be a group and let HH and KK be normal subgroups of GG. The group GG is the internal direct product of HH and KK when H∩K={e}H \cap K = \{e\} and HK=GHK = G.

Remarks

In this case, hk=khhk = kh for all h∈Hh \in H and k∈Kk \in K, because hkh−1k−1hkh^{-1}k^{-1} lies in H∩KH \cap K. Then the map (h,k)↦hk(h, k) \mapsto hk is an isomorphism from the direct product H×KH \times K to GG. Also, each element of GG is a product hkhk with h∈Hh \in H and k∈Kk \in K in exactly one way. For example, ℤ/6ℤ\mathbb{Z}/6\mathbb{Z} is the internal direct product of {0,3}\{0, 3\} and {0,2,4}\{0, 2, 4\}.

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To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptConjugate elements, definition, no evidence levelConjugate elementsdefinitionDirect product of groups, definition, no evidence levelDirect product of gro…definitionSubgroup, definition, no evidence levelSubgroupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionProduct of two subgroups, definition, no evidence levelProduct of two subgro…definitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionInternal direct product, definition, no evidence levelInternal direct produ…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Conjugate elements definition
  3. Direct product of groups definition
  4. Subgroup definition
  5. Conjugate subgroup definition
  6. Product of two subgroups definition
  7. Normal subgroup definition
  8. Internal direct product definition

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Direct_product_of_groups, Algebraic structure
About this page
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URIm2://0Q2MVQ7JMT
Kinddefinition
WritingW0 stub
MSC20E34
DefinesInternal direct product
UsesDirect product of groups, Normal subgroup, Product of two subgroups

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