Groups and Symmetry
Springer Science & Business Media, 14 mar. 2013 - 187 páginas
Groups are important because they measure symmetry. This text, designed for undergraduate mathematics students, provides a gentle introduction to the vocabulary and many of the highlights of elementary group theory. Written in an informal style, the material is divided into short sections, each of which deals with an important result or a new idea. Throughout the book, emphasis is placed on concrete examples, often geometrical in nature, so that finite rotation groups and the 17 wallpaper groups are treated in detail alongside theoretical results such as Lagrange's theorem, the Sylow theorems, and the classification theorem for finitely generated abelian groups. A novel feature at this level is a proof of the Nielsen-Schreier theorem, using groups actions on trees. There are more than 300 exercises and approximately 60 illustrations to help develop the student's intuition.
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3-cycles abelian group automorphism axis belongs to H bijection called centre CHAPTER commutator complex numbers conjugacy classes conjugate correspondence cosh cube cyclic group cyclic permutations defined denote determined diagonal dihedral group dodecahedron edges element of G elements of order equivalence classes equivalence relation example Exercise Figure finite order form a group form a subgroup free group G and G G is isomorphic gives glide reflection group G group of order hexagonal homomorphism identity element infinite integer inverse isometry isomorphic to Z2 lattice Let G Let H modulo multiplication modulo non-zero normal subgroup number of elements orbit ordered pairs orthogonal point group positive integer Proof quotient group real numbers reduced word regular tetrahedron rºs rotational symmetry group semidirect product sends sinh subgroup of G subgroup of order subset Suppose Theorem torsion coefficients translation vector vertex vertices wallpaper group x e G zero