Prove that if a and b are elements of an Abelian group G, with o(a) = m and o(b) = n, then (ab)^{mn}

Question

(a) Prove that if a and b are elements of an Abelian group G, with o(a) = m and o(b) = n,then (ab)^{mn} = e. Indicate where you use the condition that G is Abelian.

(b) With G, a, and b as in part (a), prove that o(ab) divides o(a)o(b).

(c) Give an example of an Abelian group G and elements a and b in G such that o(ab) $\neq$ o(a)o(b). Compare part (b).

Answer

a) $a^{m}=e$, $b^n=e$ , which follows $a^{mn}=e$, $b^{mn}=e$. Since G is abelian, $(ab)^{mn}=a^{mn}b^{mn}=e$.

for b i know it somthing related to gcd(m,n) and lcm(m,n) but i dont know hot to start the prove

and i think the the solution of b is the begining of the solution of c.

score 1 · Answer 1 · answered Oct 22 '16 at 18:46

1

Well, if $o(ab) = k \nmid mn$ then $mn = pk+q$ with $0 < q < k$ so $(ab) ^{mn} = ((ab)^k)^p(ab)^q = (ab)^q \ne e$

For the counterexample you can think of $o((1,1)) = 2$ while $o((0,1)) = 2 = o((1,0))$ in $\Bbb Z_2 \times \Bbb Z_2$

answered Oct 22 '16 at 18:46

cronos2

1,947

If G were not Abelian, what could you use to disprove the statement for a) – cele Sep 25 '19 at 21:50
@cele take $a = (1 2), b = (2 3)$ in $S_3$. Then $ab = (1 2 3)$ has order $3$. – cronos2 Sep 25 '19 at 22:04

Prove that if a and b are elements of an Abelian group G, with o(a) = m and o(b) = n, then (ab)^{mn}

1 Answers1