Solving Group G Show: Very Important!

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Discussion Overview

The discussion revolves around the properties of group elements, specifically focusing on the order of elements in a group G. Participants are exploring the implications of the equations related to the order of products of elements and their inverses, as well as seeking clarification on definitions and relationships between these concepts.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants assert that the order of the product of two elements, o(ab), is equal to the order of the product in reverse, o(ba).
  • Others emphasize the need for clarity on the definition of the order of an element, suggesting that if o(a) = n, then a^n = e, where e is the identity element.
  • A participant proposes a relationship involving the elements a and b, suggesting that ab can be expressed as a(ba)a^-1.
  • There is a discussion about the conditions under which an integer n exists such that ab^n = e, indicating a distinction between finite and infinite orders.
  • One participant expresses confusion about the meaning of the notation, questioning whether it implies (ab)^n = e.

Areas of Agreement / Disagreement

Participants generally agree on the definitions of the order of an element but exhibit disagreement regarding the implications of these definitions and the relationships between the orders of products of elements. The discussion remains unresolved with multiple competing views on how to proceed with the problem.

Contextual Notes

Some assumptions about the properties of group elements and their orders are not fully explored, and there are unresolved mathematical steps regarding the implications of the proposed relationships.

hashimcom
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very important...

at any group G show
i)o(ab) = o(ba)
ii) o(a) = o(cac^-1) c in G

please i want solving
 
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You need to first read the rules for this forum! What have you DONE on this problem? Where exactly did you get blocked?

To start you off, what does "o(a)" MEAN? If you know that o(a)= n, what does that tell you about a? What equation can you write?
 


NOW, o(a) means order of element a
a is element in group G
 


He asked for an equation relating all your things so you can possibly take the next step.
 


hashimcom said:
NOW, o(a) means order of element a
a is element in group G
No, that is just saying what that symbol means. What is the definition of "order of element a of group G". If I say that a, an element of group, G, has order n, what does that MEAN?
 


hashimcom said:
at any group G show
i)o(ab) = o(ba)


please i want solving

for that question i have a hent which is (( write ab=a(ba)a^-1)))
 


HallsofIvy said:
No, that is just saying what that symbol means. What is the definition of "order of element a of group G". If I say that a, an element of group, G, has order n, what does that MEAN?

it means a^n=e where n is the least integer satsfy last equations
 


hashimcom said:
it means a^n=e where n is the least integer satsfy last equations

Great now what does o(ab) mean in terms of what you just said?
 


NoMoreExams said:
Great now what does o(ab) mean in terms of what you just said?

is means if its finite there exist an integer n s.t. ab^n =e
but if else ; ther exest no integer s.t. ab^n =e

now we can write ab = a(ba)a^-1
then ...compleate
 
  • #10


hashimcom said:
is means if its finite there exist an integer n s.t. ab^n =e
but if else ; ther exest no integer s.t. ab^n =e

now we can write ab = a(ba)a^-1
then ...compleate

I thought it meant that (ab)^n = e
 

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