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There is a paper

Hickin, Kenneth Keller, Bounded HNN presentations. J. Algebra 71 (1981), no. 2, 422–434

in which on page 424 it is used "Van Dyck's theorem". The closest i could get from google is that it is some kind of third-isomorphism theorem, but could not find a formulation.

Does anybody know what is this theorem?

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  • $\begingroup$ In Hickin's paper it's going I guess about splitting into semi-direct product when something is alright with certain homomorphisms from the group into other groups. Unfortunately from what Hickin writes in his usage of this mysterious theorem, no formulation can be drawn $\endgroup$
    – Victor
    Commented Dec 1, 2013 at 10:41
  • $\begingroup$ Also, if you see first three paragraphs of Theorem 1 on page 424 of Hickin's paper, it becomes clear how poorly sometimes referees of papers can work! $\endgroup$
    – Victor
    Commented Dec 1, 2013 at 10:42

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I found several statements quickly by Googling (although they varied a bit on the van/von question).

The exact formulation varied, but basically it's just the statement that if $G$ is a group given by generators $g_i$ and relations, and there's a collection of elements $h_i$ of another group $H$ that satisfy the relations, then there's a homomorphism $\varphi:G\to H$ with $\varphi(g_i)=h_i$.

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  • $\begingroup$ Thank you, Jeremy! It is hard to beleive that Hickin used phrasing "Dyck theorem" for this fact since it would strangely match his tricks he uses in the proof; and also it's quite hard to see any projection from one group in another. But may be it's exactly what he meant $\endgroup$
    – Victor
    Commented Dec 1, 2013 at 11:45
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    $\begingroup$ @Jeremy: My understanding (from the historical study by Chandler and Magnus, Springer, 1982) is that W. van Dyck was a foounder of combinatorial group theory: Math. Ann. 20 (1882), 1-44. In particular, he helped to make rigorous the initial ideas about free groups and groups given by generators and relations, as you indicate. Those with access to MathSciNet can find somewhat more detail than Google, though of course there is no review of van Dyck's old paper as such. $\endgroup$
    – Jim Humphreys
    Commented Dec 1, 2013 at 20:12
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    $\begingroup$ Hungerford attributes a slightly stronger theorem in his algebra text to Van Dyck, see Theorem 9.5 on page 67. It says that the homomorhpism above is in fact an epimorphism. This means that group given by generators and relations is largest possible such group in a sense. $\endgroup$
    – Gregor Samsa
    Commented Dec 2, 2013 at 20:47
  • $\begingroup$ @GregorSamsa: the homomorphism constructed by Jeremy is not an epimorphism in general. Hungerford supposes that the $h_i$ are generators of $H$, which is not a hypothesis in what Jeremy wrote above. $\endgroup$
    – Bruno Stonek
    Commented Nov 15 at 7:42
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See also Theorem 2.2.1 (von Dyck's Theorem) in D.J.S. Robinson, A Course in the Theory of Groups. (My edition is the first, 1981 edition.)

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Let $X$ be a set and let $R$ be a set of reduced words on $X$. Assume that a group $G$ has the presentation $<X | R>$. If $H$ is any group generated by $X$ and satis fies the relations of $G$, i.e., $w = 1$ in $H$ for all $w \in R$, then there is a surjective group homomorphism from $G$ to $H$.

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  • $\begingroup$ That's the one I know $\endgroup$
    – Julien__
    Commented Oct 12, 2015 at 16:35
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Maybe

von Dyck W., Gruppentheoretische Studien. Math. Annal., v.20, pp.1-44

Gruppentheoretische Studien.II. Math. Annal., v.22, pp.70-108 ?

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