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I have read that free groups are the "most general" groups given some generators (from Wikipedia):

The construction of a free product is similar in spirit to the construction of a free group (the most general group that can be made from a given set of generators).

I'd like to know: is a free group the most general possible type of group? That is, are there any groups that cannot be described in group presentation (i.e. with a set of generators and a complete set of relations)? If so, what would such a group look like, and how would it relate to a free group? (Can you get that group by somehow restricting a free group? Is it more general than a free group? Are they not comparable?)

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    $\begingroup$ I think "most general" is not the right intuition. The free groups are a very special class of groups in many respects. It is true that every group has a presentation, so every group is a quotient of a free group. $\endgroup$ – Qiaochu Yuan Dec 13 '15 at 21:57
  • $\begingroup$ That wikipedia passage could be improved by saying that the free product is the "free-est" group that can be made from a given set of generators, where "free-est" means that it is as free from relators as it can possibly be. $\endgroup$ – Lee Mosher Dec 14 '15 at 16:10
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Every group has a presentation. Let $G$ be any group, and let $F$ be the free group on the underlying set of $G$. Then by the universal property of free groups, there is a homomorphism $p:F\to G$ which sends each generator to the corresponding element of $G$. Since we have a generator for each element of $G$, this homomorphism is surjective. Letting $R$ be the kernel of $p$, we then have a presentation of $G$ as the quotient of the free group $F$ by the set of relations $R$.

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