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abelian-gps

# abelian-gps - Fundamental Theorem of Abelian Groups:Algebra...

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Unformatted text preview: Fundamental Theorem of Abelian Groups :Algebra Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ ∼ squash/ 28 November, 2006 (at 21:44 ) Abstract: Seat-of-the-pants proof scribbled out one Thurs- day. Overview. We work inside a fixed group ( G , · , ε ) . Henceforth let “ cyclic group ” mean a non–one- point cyclic group . Use PoP to mean Power Of a Prime . A group G is a PoP group if # G is a PoP. If # G = p N , for a posint N and prime p , then we call G a “ p- group ” . The standing notational assumption is that G ⊃ M , F are groups and β ∈ G is a particular el- ement. I use Greek letters β ,μ,γ,ν ... to name ele- ments of groups. Our goal is to prove the following classical theorem. 1 : Fund. Theorem of Finite Abelian Groups (FToAG). Each finite abelian group G is isomorphic to a finite cartesian product of cyclic groups. The multiset of sizes of the factor groups is unique, when counted appropriately. Furthermore, if G is a p-group, then it is a finite product of cyclic p-groups. ♦ The crux for FToAG is handling the special case when G is a PoP group. Tools. Let “M ⊥ F” indicate trivial intersection, M ∩ F = { ε } ; we say that M is transverse to F . A collection { C θ } θ ∈ Θ of groups is a transverse fam- ily , if each member C θ is transverse to the subgroup generated by the other members. 2 : Fact. Suppose the gps C := { C θ } θ ∈ Θ are inside an abelian gp. Then C is a transverse family IFF the only soln to eqn Y θ ∈ Θ b θ = ε , with each b θ ∈ C θ , and all but finitely-many equal ε is every b θ = ε . If we choose to enumerate C as C 1 , C 2 ,... , then C is transverse IFF each...
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abelian-gps - Fundamental Theorem of Abelian Groups:Algebra...

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