Basis Theorem for Finite Abelian Groups

In summary, the conversation is about proving a statement in exercise O, which states that if a^{l_0}b_1^{l_1}...b_n^{l_n}=e, then a^{l_0}=b_1^{l_1}=...=b_n^{l_n}=e. The speaker proposes a method of raising both sides of the equation to the power of |b1| and concludes that (a^{l_0})=e, which can then be proven by induction. However, there is uncertainty about whether the order of a is strictly greater than the order of all b's.
  • #1
Kiwi1
108
0
I am attempting to answer the attached question. I have completed parts 1-4 and am struggling with part 5.

5. Prove that if [tex]a^{l_0}b_1^{l_1}...b_n^{l_n}=e[/tex] then [tex]a^{l_0}=b_1^{l_1}=...=b_n^{l_n}=e[/tex]

If |a|>|b1|>|b2|>...>|bn| then I could raise both sides of [tex]a^{l_0}b_1^{l_1}...b_n^{l_n}=e[/tex] to the power of |b1| and would get [tex](a^{l_l})^{|b_1|}=e[/tex] from which I conclude: [tex](a^{l_0})=e[/tex] carrying on by induction gives me the required result.

But I don't think I can prove that the order of a is STRICTLY greater than the order of all b's?
 

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  • #2
Hi,
What is needed is exercise O, which I have included in the following:

51ujk9.png
 

FAQ: Basis Theorem for Finite Abelian Groups

What is the Basis Theorem for Finite Abelian Groups?

The Basis Theorem for Finite Abelian Groups is a fundamental theorem in algebra that states that every finite abelian group can be written as a direct product of cyclic groups.

How does the Basis Theorem for Finite Abelian Groups relate to the structure of finite abelian groups?

The Basis Theorem for Finite Abelian Groups provides a way to understand the structure of finite abelian groups by breaking them down into simpler, more easily understood cyclic groups.

What is the significance of the Basis Theorem for Finite Abelian Groups?

The Basis Theorem for Finite Abelian Groups is an important result in algebra because it allows us to classify and study finite abelian groups, which have many applications in mathematics and other fields.

How is the Basis Theorem for Finite Abelian Groups proven?

The Basis Theorem for Finite Abelian Groups is typically proven using induction on the order of the group, along with other properties of finite abelian groups such as the Chinese Remainder Theorem.

Can the Basis Theorem for Finite Abelian Groups be extended to infinite abelian groups?

No, the Basis Theorem for Finite Abelian Groups only applies to finite abelian groups. However, there are similar theorems for infinite abelian groups, such as the Fundamental Theorem of Finitely Generated Abelian Groups.

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