GT22. The Fundamental Theorem of Finite Abelian Groups

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Edit for 5:45: Proof of FTFAG needs more steps as follows (thanks to Jack Shotton for the example in the comments):

Case 3: First note, if some [xi] does not contain y, then it maps isomorphically to Z/mi in G/H. One can then show that G= [xi] x Gi, where Gi is generated by the other xi's. Then the induction step finishes the proof.

Since [z] does not contain y, using the first note, this completes the general induction proof.

Abstract Algebra: We state and prove the Fundamental Theorem of Finite Abelian Groups. We apply internal direct products to Sylow subgroups in this case. Steps include showing the result for finite abelian p-groups and using the combination rule for cyclic groups with relatively prime order.

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@MathDoctorBob I'm not sure that this proof (of FTFAG) is quite right. Specifically, case 2 in the bottom right of the board at 5:45. If G = Z/p^2 x Z/p, H = <(p, 0)>, x_1 = (1, 1), x_2 = (1, 2), then x_1, x_2 is a basis for G/H, but H is contained in the subgroup generated by x_1, and that generated by x_2.

cadgenottosh

I'm a big fan! Your proof is way quicker than others!

shenanigwyns

Your welcome! My version of Uh/Um is either Now or So. It's more noticeable on the early videos, but mostly trained out by now.

MathDoctorBob

Excellent stuff. Thanks for the video.

tkcannon

Bob, your lectures are truly amazing! I have noticed that you never even say "Um.." or "Uh..". Very well executed! Thank you. Looking forward to watching all of your group theory lectures (I found this particular one while searching for explanation of Fundamental Theorem of Finite Abelian Groups).

ShamLikesCandle

Great lectures. Keep up the good work!

spramodh

In this case, the product has only one factor - Z/4. You are correct: Z/2 x Z/2 and Z/4 are not isomorphic since the first group has no element of order 4.

Here's the next thing to worry about: Z/2 x Z/3 is isomorphic to Z/6 (and then there is only one abelian group of order 6). We can combine when the factors are relatively prime.

MathDoctorBob

Hi dr., how r u ? I have the following question : show that the invariant factors of Zm + Zn are the greatest common divisors and the least common multiple, if (m, n) is greater than one and mn if (m, n) = 1 . I appreciate u r help alot .

memequbbaj

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