Abstract Huge
Abstract Huge
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Abstract Algebra: Groups and properties of their elements... PLEASE HELP?
Here's two problems I need HUGE help with:
1. If G = { g1, g2, ..., gn} is an Abelian group, show that the product p = g1g2...gn of all elements of G equals the product of the elements in G of order 2.
2.Let a and b be elements of a group. Assume that |a| = m and |b| = n, where m and n are relatively prime. If ab=ba, show |ab| = mn.
I have absolutely no idea how to show question 1, and I didn't get too far with question 2. All I knew is that since m and n are relatively prime, I should use the fact that 1 = xm + yn for some integers x and y. I couldn't figure anything out even after playing with the facts that a^m = 1 and b^n = 1.
Please help with full proofs? This material is really tough for me
1. this is easier than it looks.
suppose gi is an element NOT of order 1 or 2. then gi^-1 is some "other" element of G, and when the full product of every member of G is taken, these two elements will cancel out (we can re-arrange the elements of G in the product, to put the inverses next to each other, since G is abelian). so the only elements that won't cancel out in pairs, are the elements of order 2, and e (which makes no difference to the product).
2.first of all, if ab = ba, then (ab)^2 = abab = a(ba)b = a(ab)b = a^2b^2. in a similar vein (a rigorous proof needs induction), (ab)^n = (a^n)(b^n).
so (ab)^mn = (a^mn)(b^mn) = ((a^m)^n)((b^n)^m)) = (e^n)(e^m) = e
this at least shows |ab| divides mn.
now since (ab)^k = (a^k)(b^k), if (ab)^k = e, then both a^k = e, and b^k = e
(remember that |a| and |b| are relatively prime, so we can't have a^k = x, b^k = x^-1,
unless x = e, because if a^k = b^j, then |a^k| divides |a| and also divides |b| (since
a^k is also in ) meaning |a^k| is a common divisor of m and n, that is to say |a^k| = 1.)
thus |a| divides |ab| and |b| divides |ab|, so m and n divide |ab|, hence lcm(m,n) divides |ab|.
but m and n are relatively prime, so lcm(m,n) = mn, which divides |ab|.
(perhaps it will help to consider a group like Z12. in Z12, 6 has order 2, while 4 has order 3.
so what is the order of 10 ( = 6 + 4)?
(6 + 4) = 10
2(6 + 4) = 2(6) + 2(4) = 0 + 8 = 8
3(6 + 4) = 3(6) + 3(4) = 6 + 0 = 6
4(6 + 4) = 4(6) + 4(4) = 0 + 4 = 4
5(6 + 4) = 5(6) + 5(4) = 6 + 8 = 2
6(6 + 4) = 6(6) + 6(4) = 0 + 0 = 0, 10 has order 6 = 2*3.).



























































































