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{[itex]r \geq 1: x^r = 1[/itex]} If [itex]\theta[/itex]: G --> H is an injective group homomorphism show that, for each xEG, ord([itex]\theta(x)[/itex]) = ord(x) My answer: Please verify If [itex]\theta(x)[/itex] = {[itex]x^r: r \epsilon Z[/itex]} then ord([itex]\theta(x)[/itex]) = 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]" where [G] is an index of H in G, an integer. So order for any xEG divides order of the group. So ord([itex]\theta(x)[/itex]) = ord(x) any suggestions or changes please? thnx :)
This makes no sense. [itex]\theta(x)[/itex] is a single member of H, not a set of members of G. An "index of H in G"? It is not said here that H has to be a subset of G! Seems to me you could just use the fact that, for any injective homomorphism, [itex]\theta[/itex], [itex]\theta(x^r)= [\theta(x)]^r[/itex] and [itex]\theta(1_G)= 1_H[/itex].
i believe we have to show 2 things: i) [itex](\theta(x))^a = e'[/itex] ii) [itex]0 < b < a \implies (\theta(x))^b \neq e'[/itex]. ok so basically: If ord(x)=a then [itex]\left[ {\phi (x)} \right]^a = \left[ {\phi (x^a )} \right] = \phi (e) = e'[/itex]. Now suppose that [itex]ord\left[ {\phi (x)} \right] = b < a[/itex]. Then [itex]\left[ {\phi (x)} \right]^b = e' = \left[ {\phi (x)} \right]^a [/itex] [itex]\phi (x^b ) = \phi (x^a )[/itex] [itex]x^b = x^a[/itex] (injective) [itex]x^{a - b} = e[/itex] there seems to be a contradiction where if [itex]x^{a} = x^{b}[/itex], then [itex]\left[ {\phi (x)} \right]^b = e'[/itex] which is not what statement (ii) says. am i correct in this assumption? any ideas on how to deal with this?
When you have [itex]x^{a-b} = e[/itex], then a-b is positive since you assumed b<a. But this contradicts the definition of order. Done.