By Michael Artin

This e-book is for the honors undergraduate or introductory graduate direction. Linear algebra is tightly built-in into the textual content.

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This e-book is for the honors undergraduate or introductory graduate direction. Linear algebra is tightly built-in into the textual content.

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**Additional resources for Algebra**

**Example text**

For example, the identity elements in isomorphic groups G and G' correspond. To see this, say that the identity element 1 of G corresponds to an element E' in G Let a' be an arbitrary element of G " and let a be the corresponding element of G. By assumption, products correspond to products. Since la = a in G, it follows that E 'a' = a' in G'. In this way, one shows that E' = 1'. Another example: The orders of corresponding elements are equal. If a corresponds to a' in G', then, since the correspondence is compatible with multiplication, a' = 1 if and only if I.

Let us prove this fact by induction on n. Suppose that rows j and j + 1 are equal. 3) also have two rows equal, except when i = j or i = j + 1. When Ail has two equal rows, its determinant is zero by induction. 4) are different from zero, and d(A) = +aj1d(Ajd + aj+lld(Aj+1 d. Moreover, since the rows Rj and Rj+l are equal, it follows that Ajl = Aj+1 I and that ajl = aj+1 1. Since the signs alternate, the two terms on the right side cancel, and the determinant is zero. 4). 8) If a multiple of one row is added to an adjacent row, the determinant is unchanged.

6ii, iii) are easy to verify. 4), the fact that the integer determined by this method has the form ar + bs would not be clear at all. ) We will discuss the applications of this fact to arithmetic in Chapter 11. We now come to an important abstract example of a subgroup, the cyclic subgroup generated by an arbitrary element x of a group G. We use multiplicative notation. 7) H -- { ... , x -2 , x -) , 1,x, x, .... It is a subgroup of G-the smallest subgroup which contains x. 7) correctly, we must remember that x n is a notation for a certain element of G.