Math 5520 Class Notes – Finite Abelian Groups
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چکیده
The main goal in this section is to prove the Fundamental Theorem of Abelian Groups, which roughly speaking says that every finite, abelian group is isomorphic to a product of cyclic groups. In other words, if A is a finite, abelian group, then there are positive integers, n1, . . . , nr, so that A ∼= Z/n1Z× · · · × Z/nrZ. It’s possible to have Z/n1Z × · · · × Z/nrZ ∼= Z/m1Z × · · · × Z/msZ when (n1, . . . , nr) 6= (m1, . . .ms) and even r 6= s as the next example shows, so some care is needed to give a complete characterization of a finite, abelian groups up to isomorphism.
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