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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
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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
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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
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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?
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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.
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is said to be freely indecomposable if it is nontrivial and cannot be expressed as the free product of two nontrivial groups. Examples are:
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Obviously the free product is the coproduct in the category of groups, and with amalgamation is the
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by the elements of these subgroups, and is the âmost generalâ group having these properties.
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Universality and the corresponding notions to monoids, rings, and algebras
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this myself but thought it would be worth mentioning here, as well. -
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Any group with a nontrivial periodic normal subgroup
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