The Theory of Groups

Hans J. Zassenhaus

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486165684
• Topic: Mathematics
• Subtopic: Group Theory
• Index: Mathematics

Group theory represents one of the most fundamental elements of mathematics. Indispensable in nearly every branch of the field, concepts from the theory of groups also have important applications beyond mathematics, in such areas as quantum mechanics and crystallography.
Hans J. Zassenhaus, a pioneer in the study of group theory, has designed this useful, well-written, graduate-level text to acquaint the reader with group-theoretic methods and to demonstrate their usefulness as tools in the solution of mathematical and physical problems. Starting with an exposition of the fundamental concepts of group theory, including an investigation of axioms, the calculus of complexes, and a theorem of Frobenius, the author moves on to a detailed investigation of the concept of homomorphic mapping, along with an examination of the structure and construction of composite groups from simple components. The elements of the theory of p -groups receive a coherent treatment, and the volume concludes with an explanation of a method by which solvable factor groups may be split off from a finite group.
Many of the proofs in the text are shorter and more transparent than the usual, older ones, and a series of helpful appendixes presents material new to this edition. This material includes an account of the connections between lattice theory and group theory, and many advanced exercises illustrating both lattice-theoretical ideas and the extension of group-theoretical concepts to multiplicative domains.

APPENDIX A

FURTHER EXERCISES FOR CHAP. II

(For 10 and 11 consult Exx. 15 and 16 at the end of Chap. I.)

10. Let

be an abstract group with elements a₁, …, x₁, …. Let Σ be a system of groups such that for each member

′ of Σ there is a given an isomorphism i(

′) between

and

′. Show that the set of all one-to-one correspondences c(

′,

″, a) between any

′ and any

″ which map the element i(

′)x of

′ onto the element i(

″)ax of

″ is a groupoid

(Σ,

).

11. Let

be a groupoid. For any two units e, e′ linked by x show that ex = x = xe′. Furthermore, show that the mapping of a onto x⁻¹ax is an isomorphism between

_e and

_e′. Let Σ be the system of the groups

_e,

_e′, … attached to the units e, e′, … of

. Let

be an abstract group mapped by isomorphisms i(e), i(e′), … onto

_e,

_e′, … Show that

is isomorphic to

(Σ,

).

12. Every homomorphism h of a multiplicative domain

into another multiplicative domain defines on

the normal multiplicative congruence relation: a R(h)b if and only if ha = hb. Conversely, if R is a normal multiplicative congruence relation on

, then the residue classes form a multiplicative domain

/R according to the rule of multiplic...