Let {N i | i I| be a family of subgroups of a group G. Then

Let {N_i| i ϵ I| be a family of subgroups of a group G. Then G is the internal weak direct product of {N_i| i ϵ I} if and only if: (i) a_i,a_i = a_ia_i for all i ≠ j and a_i ϵ N_i, a_i ϵ N_i; (ii) every nonidentity element of G is uniquely a product a_i1 • • • a_in where i₁, ... , i_n are distinct elements of I and e ≠ c a_ik ϵ N_ik for each k. [Compare Theorem 8.9.]

Data From Theorem 8.9

Theorem 8.9.

Let { N_i| i ϵ I| be a family of normal subgroups of a group G. G is the internal weak direct product of the family { Ni| i ϵ I| if and only if every nonidentity element of G is a unique product a_i1a_i2 • • • a_in with i₁, ... , i_n distinct elements of I and e ≠ a_ik ϵ N_ik for each k = 1,2, ... , n.

PROOF.

Exercise. ■
There is a distinction between internal and external weak direct products. If a group G is the internal weak direct product of groups N_i, then by definition each N_i is actually a subgroup of G and G is isomorphic to the external weak direct product 

However, the external weak direct product 

does not actually contain the groups N_i, but only isomorphic copies of them (namely the L₁(N_i)- see Theorem 8.4 and Exercise 10). Practically speaking, this distinction is not very important and the adjectives "internal .. and "external .. will be omitted whenever no confusion is possible. In fact we shall use the following notation.

NOTATION.

We write  

to indicate that the group G is the internal weak direct product of the family of its subgroups {N_i|i ϵ I|.

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