By I. S. Luthar

This is often the 1st quantity of the booklet Algebra deliberate via the authors to supply enough practise in algebra to potential academics and researchers in arithmetic and comparable parts. starting with teams of symmetries of airplane configurations, it stories teams (with operators) and their homomorphisms, shows of teams through turbines and kinfolk, direct and semidirect items, Sylow's theorems, soluble, nilpotent and Abelian teams. the amount ends with Jordan's class of finite subgroups of the gang of orthogonal changes of R3. an enticing characteristic of the ebook is its richness in useful examples and instructive workouts with a spotlight at the roots of algebra in quantity thought, geometry and conception of equations

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**Example text**

B) X cannot be written as the union of two closed proper subsets. (c) Every non-empty open subset of X is dense. Proof. Exercise. Observe that when U is itself a closed subspace of X, then U is irreducible if and only if U cannot be written as the union of two closed subspaces of X different from U . When k is algebraically closed, the irreducible affine varieties correspond to prime ideals. 15. Assume that k is an algebraically closed field. Let I be a radical ideal in An . Then V (I) is irreducible if and only if I is a prime ideal.

Then every G-representation (V, m) is locally finite. Proof. Let (V, m) be an arbitrary G-representation; it suffices to show that every v ∈ V is contained in some finite-dimensional subrepresentation. Consider a basis (ei )i∈I for the k-vector space k[G], and write m(v) = i ei ⊗ vi , where each vi ∈ V , and almost all vi are zero. On the other hand, we can write ∆(ei ) = j, rij (ej ⊗ e ), where each rij ∈ k, and each of these sums is a finite sum. We now invoke 4 We will sometimes say that V is a G-representation, and not mention m explicitly; more formally, the subrepresentation corresponding to W is the pair (W, m|W ).

Then ϕ∗ is surjective because A is generated by s1 , . . , sn . If we denote its kernel by I, then A ∼ = An /I; note that I is a radical ideal because A is reduced. Let X = im ϕ ⊆ k n ; it remains to show that X = V (I). Observe that each α ∈ Y is uniquely determined by ϕ(α) ∈ k n because s1 , . . , sn generate the k-algebra A. An element x = (x1 , . . , xn ) ∈ k n , viewed as an element of homk-alg (An , k), defines an element of Y (equivalently, is contained in X = im ϕ) precisely when the elements of I vanish on x, or equivalently, when x ∈ V (I).