Abstract
We recall that the subgroup 〈S〉 of a group G generated by a set S of elements of G is the set of all products
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- 1.
This group G is the lamplighter group.
- 2.
This gives an example of a conjugation that “shrinks” a subgroup.
- 3.
This is the so called “universal property” of free abelian groups. It will be used to define freeness for nonabelian groups (Section 4.6).
- 4.
Kaplanski, Theorem 21.
- 5.
This result also holds in case A is a free abelian group of infinite rank. Using the axiom of choice, one can prove that a subgroup of A is also free, of rank at most equal to that of A (see Rotman, Theorem 10.18).
- 6.
The e i are the same as those of Theorem 4.8 taken in the reverse order.
- 7.
For this implication, assume that the subgroup theorem for free abelian groups, that we have proved for f.g. groups, holds in general; see footnote 5.
- 8.
In connection with this result it is interesting to point out the following Whitehead’s problem: is it true that if all surjective homomorphisms of an abelian group G to A with kernel Z are such that G ≃ A ⊕ Z, then A is free? This problem has been proved to be undecidable.
- 9.
Following van der Waerden we write αx.w for αx(w).
- 10.
If o(a) = o(b) = 2 the subgroup generated by a and b is dihedral; see next section, Ex. 4.6, 2 and 3.
- 11.
M. Hall, p. 105–106; P. de la Harpe, p. 23.
- 12.
P. de la Harpe, p. 26.
- 13.
A brief discussion about the definition of deficiency can be found in Macdonald, pp. 163–165.
- 14.
Chapter 7, inequality (7.29).
- 15.
We assume known the basic notions of topology.
- 16.
For a direct proof see Magnus-Karrass-Solitar, p. 42 ex;. 6, or Lyndon-Schutzen- berger, Michigan Math J. 9 (1962), p. 289.
- 17.
Posed by Max Dehn (1911).
- 18.
E.L. Post, 1945; P.S. Novikov, 1955.
- 19.
An abstract group property is a property which is preserved under isomorphism.
- 20.
We follow Macdonald, p. 167.
- 21.
Cf. Lyndon-Schupp, Theorem 4.3.
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© 2012 Springer-Verlag Italia
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Machì, A. (2012). Generators and Relations. In: Groups. UNITEXT(). Springer, Milano. https://doi.org/10.1007/978-88-470-2421-2_4
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