Direct Sum of Groups - are there 2 types of direct sums?

Question

This has been a point of confusion for quite some time now for me.

I have come across two different definitions of the direct sum on groups, each with a different notation for the symbol. These two definitions are given here:

1. https://en.wikipedia.org/wiki/Direct_sum_of_groups
2. https://en.wikipedia.org/wiki/Direct_sum#Direct_sum_of_abelian_groups

Clearly these two definitions cannot be the same, as one says the set of the direct sum group is the cartesian product of the two sets, wheras the other one says, it is the set generated by the two normal subgroups.

So which one is correct, are both of these 'equivalent', and how do we know which definition of direct sum to take in a given context?

Answer 1

Yes, they're equivalent. The cartesian product one is the external direct sum; the normal subgroup one is the internal direct sum. For a proof they are isomorphic, see Theorem 9.6 of Gallian's, "Contemporary Abstract Algebra (Eighth Edition)".