What is the Factor Group (ℤ4 ⊕ ℤ12)/＜(2,2)＞ Isomorphic To? (Group Theory)
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- Опубліковано 8 тра 2024
- In Group Theory, the direct product ℤ4 ⊕ ℤ12 is a non-cyclic group of order 4*12=48. The cyclic subgroup H=＜(2,2)＞={(0,0),(2,2),(0,4),(2,6),(0,8),(2,10)} has order 6. Therefore, by Lagrange's Theorem, the factor group (a.k.a. quotient group) (ℤ4 ⊕ ℤ12)/＜(2,2)＞ has order 48/6 = 8. Its elements are cosets. It is also an Abelian group. The possible isomorphism classes are the external direct products ℤ8 , ℤ4 ⊕ ℤ2, and ℤ2 ⊕ ℤ2 ⊕ ℤ2. Which one is the right answer? This is an interesting problem in Abstract Algebra.
#GroupTheory #AbstractAlgebra #FactorGroup #QuotientGroup #DirectProduct
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Great lesson. I am a mathematician too, but I do complex geometry on manifolds. I have an infinite admiration for algebraists. Thanks much!
You're welcome! I'm actually not an algebraist. I just happen to teach abstract algebra. My work was in dynamical systems. So glad you liked the video!
So is this supposed to be another form of this cyclic group? but can this apply the lcm thing like Z3 + Z9 OR Z2 + Z3?
Since (ℤ4 ⊕ ℤ12)/＜(2,2)＞ is isomorphic to ℤ2 ⊕ ℤ4, you can apply the lcm idea to the coset representatives to get their orders. Does that answer your question?
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@@arcturusgd I'll try if I have time, though I'm pretty busy right now.