Internal direct product

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This article describes a product notion for groups. See other related product notions for groups.


This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Definition

Definition with symbols (for two subgroups)

A group G is termed the internal direct product of two subgroups N_1 and N_2 if both the following conditions are satisfied:

Equivalently G is the internal direct product of N_1 and N_2 if both the following conditions are satisfied:

  • Every element of N_1 commutes with every element of N_2. In other words, N_1 is contained in the centralizer of N_2.
  • N_1 and N_2 are lattice complements, that is, they intersect trivially and together they generate G, i.e., the join of subgroups \langle N_1,N_2 \rangle is equal to G.

The two subgroups N_1 and N_2 are termed direct factors of G.

Definition with symbols (for arbitrary family of subgroups)

A group G is termed the internal direct product of subgroups N_i, i \in I, if the following three conditions are satisfied:

  1. Each N_i is a normal subgroup of G
  2. The N_is generate G
  3. Each N_i intersects trivially the subgroup generated by the other N_js. Equivalently, if g_1g_r\dots g_r = e where g_l \in N_{j_l} with all j_l distinct, then each g_l = e.

Equivalently, G is the internal direct product of the N_is if the following two conditions are satisfied:

  1. Every element of N_i commutes with every element of N_j for i \ne j
  2. Each N_i is a lattice complement to the subgroup generated by the remaining N_js

Equivalence of definitions

Further information: equivalence of definitions of internal direct product

Equivalence with the external direct product

Further information: equivalence of internal and external direct product

It can be proved that if G is an internal direct product of subgroups N_1 and N_2, then G is isomorphic to the external direct product N_1 × N_2 via the isomorphism that sends a pair (a,b) from N_1 \times N_2 to the product ab in G. Conversely, given an external direct product N_1 \times N_2, we can find subgroups isomorphic to N_1 and N_2 in the external direct product such that it is the internal direct product of those subgroups.

For infinite collections of subgroups, the internal direct product does not coincide with the external direct product -- instead, it coincides with the notion of restricted external direct product.

Relation with other properties

Weaker product notions

Related subgroup properties