
#1
Jan1208, 06:10 PM

P: 39

Hi i have completed the answer to this question. Just need your verification on whether it's completely correct or not:
Question: If G is a group and xEG we define the order ord(x) by: ord(x) = min{[latex]r \geq 1: x^r = 1[/latex]} If [latex]\theta[/latex]: G > H is an injective group homomorphism show that, for each xEG, ord([latex]\theta(x)[/latex]) = ord(x) My answer: Please verify If [latex]\theta(x)[/latex] = {[latex]x^r: r \epsilon Z[/latex]} then ord([latex]\theta(x)[/latex]) = ord(x). For any integer r, we have x^r = e (or 1) if and only if ord(x) divides r. In general the order of any subgroup of G divides the order of G. If H is a subgroup of G then "ord (G) / ord(H) = [G:H]" where [G:H] is an index of H in G, an integer. So order for any xEG divides order of the group. So ord([latex]\theta(x)[/latex]) = ord(x) any suggestions or changes please? thnx :) 



#2
Jan1208, 08:32 PM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,877





#3
Jan1308, 07:29 AM

P: 39

i believe we have to show 2 things:
i) [latex](\theta(x))^a = e'[/latex] ii) [latex]0 < b < a \implies (\theta(x))^b \neq e'[/latex]. ok so basically: If ord(x)=a then [latex]\left[ {\phi (x)} \right]^a = \left[ {\phi (x^a )} \right] = \phi (e) = e'[/latex]. Now suppose that [latex]ord\left[ {\phi (x)} \right] = b < a[/latex]. Then [latex]\left[ {\phi (x)} \right]^b = e' = \left[ {\phi (x)} \right]^a [/latex] [latex]\phi (x^b ) = \phi (x^a )[/latex] [latex]x^b = x^a[/latex] (injective) [latex]x^{a  b} = e[/latex] there seems to be a contradiction where if [latex]x^{a} = x^{b}[/latex], then [latex]\left[ {\phi (x)} \right]^b = e'[/latex] which is not what statement (ii) says. am i correct in this assumption? any ideas on how to deal with this? 



#4
Jan1608, 09:12 PM

P: 49

Orders of Groups
When you have [latex]x^{ab} = e[/latex], then ab is positive since you assumed b<a. But this contradicts the definition of order. Done.



Register to reply 
Related Discussions  
# of diffraction orders seen???  Introductory Physics Homework  3  
Orders of a product?  Calculus & Beyond Homework  0  
Orders and Degrees  Differential Equations  6  
Different orders of dimensions  Beyond the Standard Model  5  
Wallpaper Groups, Free Groups, and Trees  Introductory Physics Homework  13 