Fundamental Theorem of Finitely Generated Abelian Groups -- Abstract Algebra Examples 16
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- Опубліковано 20 кві 2023
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Great teacher, thank you Michael for this great educational project ... wish you the best !
Are the four possible abelian groups of order 450 not isomorphic to each other? I see each one can be isomorphic to multiple different groups by combining groups whose p values are relatively prime. But Z2 x Z9 x Z25 is NOT isomorphic to Z2 x Z2 x Z3 x Z25?
First question: they are not isomorphic
Second question: the groups in your question are not isomorphic because they have different order (the first one has order 450, the second one has order 300).
the fundamental theorem of me getting completely lost in the first minute of learning about finitely generated abelian groups...
all you need now is to add prime number modulus theory and i have found the fundamental theorem of being completely incapable of comprehending it...
who would have thought getting an astrophysics degree would be easier...
Doesn't the theorem at the end apply to any finite non-abelian p-group?
A slick way to prove 9:31 is to go by contrapositive: if H and K are nonisomorphic then they have different "fundamental forms", and then their products with the "fundamental form" of G are still different so G×H and G×K are nonisomorphic
What does ""fundamental form" mean here? Also, why does your proof only work for finitely generated abelian groups, but not for others?
@@bjornfeuerbacher5514 By "fundamental form" I mean "canonical form given by the fundamental theorem of finitely generated abelian groups" (so we need the abelian groups to be finitely generated for the theorem to apply)