IAAWA Groups

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Chapter 8 Groups

We now turn to groups, which are an algebraic structure with a single operation. As we shall see, we can construct a group out of a ring by taking only the additive operation, or by taking the multiplicative operation and restricted to the set of units. For example, the integers $\mathbb Z$ under addition and the invertible $2 \times 2$ matrices under multiplication both form examples of groups. In addition, groups arise from the symmetries of any geometric object.

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.