Abelian group order divides 1cm I want to prove the following: Let G be an abelian group and let a1,⋯,ar∈G then |a1⋯ar| (the order of a1⋯ar) divides 1cm(|a1|,…,|ar|) So far I have shown that (a1,…,ar)1cm(|a1|,…,|ar|)=1 but I dont

It does imply the desired result. Note that if an=1, then |a| divides n. Simply write n=|a|q+r for 0≤r&lt;|a|, and then 1=an=a|a|q⋅ar=1q⋅ar=ar, and so since r&lt;|a| and ar=1, we have r=0.

Resolved: "Abelian group order divides 1cm I want to..."

College
Algebra
Abstract algebra

Kamari Simon

Answered question

2022-02-15

Abelian group order divides 1cm
I want to prove the following:
Let G be an abelian group and let

a_{1}, \dots, a_{r} \in G

then

| a_{1} \dots a_{r} |

(the order of

a_{1} \dots a_{r}

) divides

1 c m (| a_{1} |, \dots, | a_{r} |)

So far I have shown that

{(a_{1}, \dots, a_{r})}^{1 c m (| a_{1} |, \dots, | a_{r} |)} = 1

but I dont

Answer & Explanation

Anahi Pearson

Beginner2022-02-16Added 16 answers

It does imply the desired result. Note that if

a^{n} = 1

, then |a| divides n. Simply write

n = | a | q + r for 0 \leq r < | a |

, and then

1 = a^{n} = a^{| a | q} \cdot a^{r} = 1^{q} \cdot a^{r} = a^{r}

,
and so since

r < | a | and a^{r} = 1

, we have

r = 0

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Abstract algebra

How to find out the mirror image of a point?
Generators of a free group
If G is a free group generated by n elements, is it possible to find an isomorphism of G with a free group generated by n-1 (or any fewer number) of elements?
How many 3/4 Are in 1
Convert 10 meters to feet. Round your answer to the nearest tenth
6. Reduce the following matrix to reduced row echelon form:
$C equals open parentheses table row 1 cell negative 1 end cell cell 1 half end cell cell 1 half end cell row 0 1 cell 1 half end cell cell negative 1 half end cell row 0 0 1 0 row 0 0 0 0 end table close parentheses$
Let v be a vector over a field F with zero vector 0 and let s,T be a substance of V .then which of the following statements are false
Describe Aut(Zp), the automorphism group of the cyclic group Zp where p is prime. In particular find the order of this group. (Hint: A generator must map to another generator)

Didn't find what you were looking for?