Simple & Solvable Groups (12/2)

1. Simple & Solvable Groups (12/2) • We assume that our groups are finite. • Definition. A group G is called simple if it contains no proper non-trivial normal subgroups. • Theorem. An abeliangroup G is simple if and only if____________________________. • Question: Do there exist non-abelian simple groups? • Examples: Dnis not simple for all n > 2. (Why?) • Examples: Snis not simple for all n > 2. (Why?) • Example: A4 is not simple. (Why?)

2. But, A5Is Simple! • Very Handy Lemma. Let H be a normal subgroup of G and let x be any element of G. If |x| and |G / H | are relatively prime, then x H. • Proof. _____________________________________.(Question: Where did the normality of H come in??) • Theorem. A5 is simple. • Proof. ______________________________________. • Theorem. Anis simple for all n > 4.

3. Solvable Groups • Definition. A nested sequence of subgroups of G, {e} = H0 ≤ H1 ≤H2 ≤ …Hk-1 ≤ Hk = G ,is called a sub-normal seriesif each Hiis normal in Hi+1. • Example. If K = {(1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}, then {(1)} ≤ (1 2)(3 4) ≤ K ≤ A4 ≤ S4is a sub-normal series of S4. • Definition. A group G is called solvableif it contains a subnormal series in which each of the corresponding factor groups is abelian. • Example. Check that the above series shows that S4 is solvable.

4. SnIs Not Solvable for n > 4 • For all n > 4, the simplicity of An prohibits Snfrom possessing a sub-normal series with “abelian factors”, hence rendering them unsolvable. • Theorem. There exist algorithms for solving (exactly, in terms of radicals) an arbitrary polynomial with coefficients in Zprovided the degree of the polynomial is 1, 2, 3 or 4. • Theorem. There cannot exist an algorithm for solving (exactly, in terms of radicals) an arbitrary polynomial with coefficients in Z if its degree is greater than 4. • Proof. This follows from the non-solvability of Sn. This area of Abstract Algebra is called Galois Theory and is very beautiful! (See MA 320 – Abstract Algebra II.)