Resource:Example-based coverage articles

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This page lists articles that cover a whole range of definitions, proofs and results in group theory, by developing these concepts as examples for some other, related, subject.

For each of these articles, we briefly describe the other subject or context in which the definitions and results are discussed. We also give a list of important definitions, proofs and results that are covered.

Elementary group theory

This may be of particular interest to beginners, and to people interested in educating beginners through examples.

Group theory via geometry

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This develops many of the basic notions of group theory by looking at various subgroups of the group of transformations of the Euclidean plane, and the relation between them

Concepts revised: group, subgroup, direct product, index of a subgroup, homomorphism of groups, normal subgroup, permutable complements, semidirect product, retract, kernel, quotient group

Ideas touched upon: [Intermediate subgroup condition]], quotient-transitive subgroup property, intersection-closed subgroup property

Group theory via permutations

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This develops many of the basic notions of group theory by looking at various subgroups of the symmetric group over a set (we consider finite as well as infinite sets):

Concepts revised: group, subgroup, automorphism of a group, inner automorphism, conjugate subgroups, normal subgroup, characteristic subgroup, direct product, semidirect product, retract, Abelian group, cyclic group, maximal subgroup, transitive group action, faithful group action, primitive group action

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Group theory via number theory

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This starts off by looking at the particular case of modular arithmetic, and some groups that arise as the additive and multiplicative groups modulo integers. We then proceed to look at certain other groups, which arise in the context of rings of integers. The focus is mainly on Abelian group theory.

Concepts revised: group, Abelian group, cyclic group

Results revised: structure theorem for finitely generated Abelian groups

Group theory via topology

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This starts off by viewing the group of loops, upto homotopy, under loop concatenation, the so-called fundamental group of a topological space. Many related groups are developed, and basic notions of group theory are revised in the process:

Concepts revised: group, Abelian group, cyclic group, trivial group, inner automorphism, normal subgroup, direct product