Orders of Groups

smoothman

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 :)

HallsofIvy

Quote by smoothman

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).

This makes no sense. [itex]\theta(x)[/itex] is a single member of H, not a set of members of G.

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.

An "index of H in G"? It is not said here that H has to be a subset of G!

So order for any xEG divides order of the group. So ord([latex]\theta(x)[/latex]) = ord(x)

any suggestions or changes please? thnx :)

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].

smoothman

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?

masnevets

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

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