By Larry C. Grove

ISBN-10: 0080874290

ISBN-13: 9780080874296

ISBN-10: 0123046203

ISBN-13: 9780123046208

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**Extra info for Algebra**

**Example text**

If n # 4, then the alternating group A , is simple. Proof. A , , A , , and A , are simple since their orders are 1, 1, and 3. Suppose n 2 5 and take H d A,, H # 1. 5 it will suffice to show that there is a 3-cycle in H in order to conclude that H = A,. 7). Then o is a product of k disjoint p-cycles for some k. If p = 3 and k = 1 we are finished. Otherwise there are four cases to consider. Case 1: p > 3. Say CT = (alu2. . u p ).... Then a(alu2a3)Q-'(a~a3a2) = (a2a3a4)(ala3a2) = (ala4a2) H. Case 2: p = 3 and k > 1.

Suppose f : G + A is a homomorphism, with A abelian. Since every commutator is in the kernel of j' we have G' I k e r f . Thus if xG' = yG' or y - ' x E G', then f ( y - ' x ) = 1, or f ( x ) = f(y), and we may define 24 I Group g(xG’) = f ( x ) . If also g1: U -+ A is a homomorphism such that glE = f, then glE = gE and so g1 = g since E is an epimorphism. It follows that (V, E ) is a universal pair for G. 3. (1) Find G‘ if G = S, , S,, or A , . (2) If G’ I H I G show that H 4G. 2 to show that if K U G, then K‘ U G.

Proof. 1. 1. If S = { 1 ) and N = { 1,2,3,. } is the semigroup of natural numbers under addition, verify that N is a free semigroup on S, with j ( 1 ) = 1. 2. If S is any nonempty set then a free semigroup X on Sexists. Proof. finitely many copies of S . 5 bk) = * * * , a m ,b1, * * - 9 bk), then it is immediate that the operation is associative, so S is a semigroup. Take -+ X to be the inclusion map, j ( x ) = x. If Y is any semigroup and k: S Y a function define f : X -+ Y by setting f ( a , , ..

### Algebra by Larry C. Grove

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