Abstract
Associated with a set X is the free group F X spanned by X and described as follows. Consider an alphabet formed by letters x and x −1, where x ∈ X. On the set of all words of this alphabet consider the smallest equivalence relation “ = ” that identifies two words obtained from each other by inserting or deleting any number of copies of xx −1 or x −1 x (or both) at the beginning, or at the end, or between any two sequential letters. By definition, the elements of the free group F X are the equivalence classes of words with respect to this equivalence. The composition is the concatenation of words: \(x1x2\,\ldots \,xk \cdot y1y2\,\ldots \,ym\stackrel{\text{def}}{=}x1x2\,\ldots \,xky1y2\,\ldots \,ym\).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Including the empty word \(\varnothing \).
- 2.
That is, the intersection of all normal subgroups containing R.
- 3.
This is a part of the famous undecidability of the word problem proved by Pyotr Novikov in 1955.
- 4.
With vertices at the barycenter of the face, at the barycenters of its edges, and at its vertices.
- 5.
Note that we get a new explanation for the identity \(\vert \mathop{\mathrm{O}}\nolimits _{\Phi }\vert = N = 6 \cdot (\text{number of faces})\).
- 6.
That is, in the opposite order to that in which the reflections are made.
- 7.
For instance by a geodesic cut out of the sphere by a plane passing through the center of the sphere.
- 8.
That is, arcs of a great circle cut out of the sphere by the plane passing through a, b, and the center of sphere.
- 9.
This always can be achieved by a small perturbation of b, because there are finitely many geodesics passing through a and some vertex of the triangulation.
- 10.
Equivalently, the hyperplane \(\mathbb{A}^{n} \subset \mathbb{R}^{n+1}\) is given by the equation x 0 + x 1 + ⋯ + x n = 1.
- 11.
See Problem 10.9 on p. 249.
- 12.
See Sect. 6.5.5 on p. 148.
- 13.
Which is cut out of the sphere by the reflection plane \(\pi _{i_{\nu }}\).
- 14.
That is, the shortest of two arcs of the great circle cut out of \(S^{n-1} \subset \mathbb{A}^{n}\) by the 2-dimensional plane passing through a, b, and the center of sphere.
- 15.
See Sect. 9.2 on p. 208.
- 16.
See Sect. 1.4 on p. 13.
- 17.
Or just length for short.
- 18.
Consisting of the identity and three pairs of disjoint transpositions of cyclic type
.
- 19.
The Mathieu groups M 11, M 12, M 22, M 23, M 24, but not M 10 are among them (see Problem 12.34 on p. 306).
- 20.
However, \(A_{3} \simeq \mathbb{Z}/(3)\) is also simple.
- 21.
Such as \(\mathrm{PSL}_{n}(\mathbb{F}_{q})\). Explicit definitions and classifying theorems for these groups can be found in textbooks on linear algebraic and/or arithmetic groups, e.g. Linear Algebraic Groups, by James E. Humphreys [Hu].
- 22.
The final part of the story is expounded in a six-volume manuscript [GLS].
- 23.
The symbol ⋊ should serve as a reminder that \(N\lhd \,N\mathop{ \rtimes }\nolimits H\).
- 24.
See Example 12.14 on p. 295.
- 25.
See Theorem 12.3 on p. 300.
- 26.
See Example 12.14 on p. 295.
- 27.
See Example 10.13 on p. 247.
- 28.
See Problem 10.13 on p. 250.
- 29.
For example, every subgroup of index 2 is normal, every subgroup of index 3 in a group of odd order is normal, etc.
References
Bernoulli, J.: Ars conjectandi, opus posthumum. Accedit Tractatus de seriebus infinitis, et epistola gallicé scripta de ludo pilae reticularis. Basileae, impensis Thurnisiorum, fratrum (1713).
Borevich, Z. I., Shafarevich, I. R.: Number Theory. Academic Press, New York (1966).
Danilov, V. I., Shokurov, V. V.: “Algebraic Curves, Algebraic Manifolds and Schemes.” In Encyclopedia of Mathematical Sciences. Springer, Heidelberg (1994).
Francis, G. K.: A Topological Picturebook. Springer, Heidelberg (1987).
Gorenstein, D., Lyons, R. Solomon, R.: The Classification of the Finite Simple Groups. Mathematical Surveys and Monographs 40, vols. 1–6. AMS Providence, R.I. (1994–2005).
Humphreys, J. E.: Linear Algebraic Groups. Springer, Heidelberg (1975).
Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory. Springer, Heidelberg (1990).
Mumford, D.: Tata Lectures on Theta I. Progress in Math, vol. 23, Birkhäuser (1983).
Serre, J.-P.: Lie Groups and Lie Algebras: 1964 Lectures Given at Harvard University. Lecture Notes in Mathematics series 2, vol. 1500, Springer (1965).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing AG
About this chapter
Cite this chapter
Gorodentsev, A.L. (2016). Descriptions of Groups. In: Algebra I. Springer, Cham. https://doi.org/10.1007/978-3-319-45285-2_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-45285-2_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-45284-5
Online ISBN: 978-3-319-45285-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)