Abstract Algebra | Definition of a Group and Basic Examples

[@georgesadler7830]

Professor Penn, thank you for an awesome introduction to Abstract Algebra. I really like Abstract Algebra. I took this class at the University of Maryland, College Park in the early 1990's.

[@grihabor]

Could you please fix the order of videos in "Abstract Algebra: The basics of groups playlist"? Thank you for your work!

[@valeriereid2337]

Excellent explanation.

[@PunmasterSTP]

I saw some Z’s, but I certainly didn’t catch them! Thanks for another entertaining ride.

[@mentormaths9058]

Kindly make more videos on advance abstract algebra of Ms mathematics

[@harshuhh]

Thanks for the vids. I'm having trouble proving (Zn, +) is closed.. it seems trivial but the examples I've seen aren't very clear. Looking swole btw... physique and math goals right there

[@RexxSchneider]

The fundamental theorem of division states that for all x, n ∈ N, there exists q, r ∈ N such that x = n.q + r and 0 ≤ r < n. This also holds if x, q ∈ ℤ. Taking the first condition (mod n), we see that x ≡ r (mod n) and that there are only n equivalence classes (mod n) which my be taken to be the values of r such that 0 ≤ r < n, and those make up the elements of Zn. Let x1, x2 ∈ Z for the moment. Then setting x1 = n.q1 + r1 and x2 = n.q2 + r2, we have x1+x2 = n.(q1+q2) + (r1+r2). That gives (x1+x2) ≡ (r1+r2) (mod n) ≡ r (mod n) because (r1+r2) can be set equal to x and therefore must be equal to n.q + r where 0 ≤ r < n. If we now confine x1, x2 ∈ Zn, we show that the sum of any two elements of Zn is an element of Zn.

[@prbprb2]

Why do we care much about associativity? Why study loops more as opposed to groups? When working with real data, generally the order of operations is implied.... Can someone discuss with me?

[@natepolidoro4565]

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