By Mark Steinberger

The purpose of this publication is to introduce readers to algebra from some degree of view that stresses examples and class. every time attainable, the most theorems are handled as instruments which may be used to build and study particular varieties of teams, earrings, fields, modules, and so on. pattern structures and classifications are given in either textual content and workouts.

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**Sample text**

We write Zn for the set of equivalence classes of integers modulo n, and write m ∈ Zn for the equivalence class containing m ∈ Z. 3 We shall also use 0 to denote 0. Zn is called the group of integers modulo n. 4. There are exactly n elements in Zn , represented by the elements 0 = 0, 1, . . , n − 1. Proof The elements listed are precisely the classes r for 0 ≤ r < n. For m ∈ Z, the Euclidean algorithm provides an equation m = qn + r, with 0 ≤ r < n. But then m = r, and hence m is in the stated list of elements.

We may then deduce in Chapter 5 that Sn is not what’s known as a solvable group for n ≥ 5. The non-solvability of Sn will allow us, in Chapter 11, to prove Galois’s famous theorem that there is no formula for solving polynomials of degree ≥ 5. This should be enough to arouse one’s interest in the symmetric groups. We also introduce a very important concept in group theory: that of the set G/H of left cosets of a subgroup H in G. One of the ﬁrst consequences of the study of cosets is Lagrange’s Theorem: if G is ﬁnite, then the order of any subgroup of G must divide the order of G.

Recall that one-to-one functions are sometimes called injective functions, or injections, and that onto functions are known as surjective functions, or surjections. Also, a function which is both one-to-one and onto is called bijective. Clearly, an understanding of homomorphisms will contribute to our understanding of isomorphisms, but in fact, homomorphisms will turn out to say a lot more about the CHAPTER 2. GROUPS: BASIC DEFINITIONS AND EXAMPLES 30 structure of a group than may be apparent at this point.