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3: Groups

Last updated

Sep 4, 2021
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- 2.8: Sage Exercises
- 3.4: Reading Questions

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We begin our study of algebraic structures by investigating sets associated with single operations that satisfy certain reasonable axioms; that is, we want to define an operation on a set in a way that will generalize such familiar structures as the integers ${\mathbb Z}$ together with the single operation of addition, or invertible $2 \times 2$ matrices together with the single operation of matrix multiplication. The integers and the $2 \times 2$ matrices, together with their respective single operations, are examples of algebraic structures known as groups.

The theory of groups occupies a central position in mathematics. Modern group theory arose from an attempt to find the roots of a polynomial in terms of its coefficients. Groups now play a central role in such areas as coding theory, counting, and the study of symmetries; many areas of biology, chemistry, and physics have benefited from group theory.

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