Book•
A Course in the Theory of Groups
19 Oct 2011-
TL;DR: A detailed introduction to the theory of groups: finite and infinite; commutative and non-commutative is given in this article, where the reader is provided with only a basic knowledge of modern algebra.
Abstract: This is a detailed introduction to the theory of groups: finite and infinite; commutative and non-commutative. Presupposing only a basic knowledge of modern algebra, it introduces the reader to the different branches of group theory and its principal accomplishments.
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04 Nov 2010
TL;DR: Theorem: Finitely Generated Amenable Groups + Local Embeddability and Sofic Groups + Uniform Structures + Complements of Functional Analysis + Ultrafilters = 6.
Abstract: Cellular Automata.- Residually Finite Groups.- Surjunctive Groups.- Amenable Groups.- The Garden of Eden Theorem.- Finitely Generated Amenable Groups.- Local Embeddability and Sofic Groups.- Linear Cellular Automata.- Nets and the Tychonoff Product Theorem.- Uniform Structures.- Symmetric Groups.- Free Groups.- Inductive Limits and Projective Limits of Groups.- The Banach-Alaoglu Theorem.- The Markov-Kakutani Fixed Point Theorem.- The Hall Harem Theorem.- Complements of Functional Analysis.- Ultrafilters.
344 citations
TL;DR: In this paper, the authors showed that two Feistel permutations are sufficient together with initial and final pairwise independent permutations for pseudorandom functions with small input-length and provided a framework in which similar constructions may be brought up and their security can be easily proved.
Abstract: Luby and Rackoff [26] showed a method for constructing a pseudorandom permutation from a pseudorandom function. The method is based on composing four (or three for weakened security) so-called Feistel permutations, each of which requires the evaluation of a pseudorandom function. We reduce somewhat the complexity of the construction and simplify its proof of security by showing that two Feistel permutations are sufficient together with initial and final pairwise independent permutations. The revised construction and proof provide a framework in which similar constructions may be brought up and their security can be easily proved. We demonstrate this by presenting some additional adjustments of the construction that achieve the following:
? Reduce the success probability of the adversary.
? Provide a construction of pseudorandom permutations with large input-length using pseudorandom functions with small input-length.
317 citations
TL;DR: In this paper, it was shown that if G and H are two non-abelian finite groups such that Γ G ≅ Γ H, then | G | = | H |, then H is nilpotent.
304 citations
TL;DR: In this paper, a subgroup called c-normal in a group is defined, where the maximal normal subgroup of the group is the subgroup contained in the normal subgroups.
290 citations
TL;DR: The model theory of existentially closed difference fields has been studied in this article, where it is shown that an arbitrary formula may be reduced into one-dimensional ones, and analyzed the possible internal structures on the onedimensional formulas when the characteristic is 0.
Abstract: A difference field is a field with a distinguished automorphism σ. This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory on formulas, and in particular on difference equations. We show that an arbitrary formula may be reduced into one-dimensional ones, and analyze the possible internal structures on the one-dimensional formulas when the characteristic is 0.
261 citations
Cites background from "A Course in the Theory of Groups"
...11 in [28]) and therefore that [G,G] is definable....
[...]