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Extra info for Algebra I

Example text

For example, the composition factors of Z6 are Z2 and Z3 . 6 Example: Fundamental theorem of arithmetic Let n be a natural number with n ≥ 2. 57 Theorem (Fundamental theorem of arithmetic). The number n is a product of prime numbers and these prime numbers are uniquely determined up to order. Here is a proof of this well-known theorem using the Jordan-H¨older theorem. The group Zn is finite, so it has a composition series (G0 , G1 , . . , Gr ). 4). By repeated use of Lagrange’s theorem we have n = p0 p1 · · · pr−1 .

6) where the orders of the simple groups turn out to be the prime factors of n. Therefore, this way of associating to a given finite group a list of simple groups can be viewed as a generalization of the associating to a given natural number its list of prime factors. Just as we regard prime numbers as building blocks of all numbers, we can regard simple groups as building blocks of all finite groups. Due to the pivotal role finite simple groups play, much effort has been expended in their determination, and this is now complete: Theorem (Classification of finite simple groups).

2). If g ∈ G, then g ∈ ker π ⇐⇒ π(g) = N ⇐⇒ gN = N ⇐⇒ g ∈ N, so ker π = N , as claimed. 6 Homomorphism is injective iff kernel is trivial Let ϕ : G → G be a homomorphism. Theorem. ϕ is injective if and only if ker ϕ = {e}. Proof. Assume that ϕ is injective. Let k ∈ ker ϕ. Then ϕ(k) = e . 3. So ϕ(k) = ϕ(e) and injectivity of ϕ gives k = e. This shows that ker ϕ ⊆ {e}. Since a kernel is a subgroup, the other inclusion is immediate. Now assume that ker ϕ = {e}. Let x, y ∈ G and assume that ϕ(x) = ϕ(y).

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Algebra I by Randall R. Holmes


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