Talk:Free product

Free product with amalgamation discussion

I think the discussion of free products with amalgamation would be benefited by at least a brief argument for the universal property of the construction. (In other words, a bit more detailed discussion into why it's the pushout in the category of groups.) If I have time, I will try to get around to this myself but thought it would be worth mentioning here, as well. - mathemajor (talk) 14:17, 15 November 2010 (UTC)

The article only defines free product, and then goes on to talk about amalgamated free products without saying what they are, nor giving a link. Selinger (talk) 03:47, 17 March 2023 (UTC)

Universality and the corresponding notions to monoids, rings, and algebras

Obviously the free product is the coproduct in the category of groups, and with amalgamation is the pushout. This is also true in the category Mon of monoids, as well as in Ring and Alg_k for k a field, or even a commutative ring. I think we should expand a section on universality in the category of groups and add a section larger than 3 lines on the free product of other algebraic structures. I'm thoroughly confused regarding the free product of rings or algebras, which is briefly mentioned over at the article Tensor product of R-algebras. --Daviddwd (talk) 21:56, 27 August 2014 (UTC)

Freely indecomposable groups

A group G is said to be freely indecomposable if it is nontrivial and cannot be expressed as the free product of two nontrivial groups. Examples are:

• Any nontrivial finite group
• Any nontrivial abelian group
• Any group with a nontrivial center
• Any group with a nontrivial periodic normal subgroup GeoffreyT2000 (talk) 04:24, 28 February 2015 (UTC)

Precision about "most general"

This article begins as follows:

In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties.

Does that mean simply that all groups having those properties are homomorphic images of this one? Is this in some sense the smallest group that has all of those as homomorphic images? Michael Hardy (talk) 20:34, 21 March 2019 (UTC)

I have tried to clarify this sentence. I am not proud of the style, so feel free to improve it. By the way, I have replaced "most general" by the correct technical term, which is "universal". I'll link it in a next edit D.Lazard (talk) 21:57, 21 March 2019 (UTC)