Recent content by smoothman

ok I've managed to solve the other 2 questions. here is my final one: (1) If G is a group and n \geq 1 , define G(n) = { x E G: ord(x) = n} (2) If G \cong H show that, for all n \geq 1 , |G(n)| = |H(n)|. (3) Deduce that, C_3 X C_3 is not \cong C_9. Is it true that C_3 X C_5...

i also understand where i have gone wrong :) thnx

thnx dodo. i agree completely with what you said. on my final presentation, i have fixed all the mistakes in notation to make it clear for the teacher. i appreciate everything u said, which is correct :) thnx for the help.

i understand exactly what you mean and i will take care of the notation in future. just for the purposes of this question i have completed the associtivity and existence of inverse element using the same notation as before.. is this correct: associativity: (g, h) * [(g', h') * (g''...

i believe we have to show 2 things: i) (\theta(x))^a = e' ii) 0 < b < a \implies (\theta(x))^b \neq e'. ok so basically: If ord(x)=a then \left[ {\phi (x)} \right]^a = \left[ {\phi (x^a )} \right] = \phi (e) = e'. Now suppose that ord\left[ {\phi (x)} \right] = b < a. Then \left[...

ok: so closure is satisfied because we define a binary operation * such that *:G\times G\mapsto G, so the group is closed. existence of identity element is satisfied because if we choose (e,E) where e is identity element in G and E is identity element in H. Then (g,h)*(e,E) = (ge,hE)=(g,h)...

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{r \geq 1: x^r = 1} If \theta: G --> H is an injective group homomorphism show that, for...

yes sorry, GXH = G x H... i understand <some property> for 1 and 2 are "." and "*" respectively.. where these are the identity elements. "o" applies to (3). but what do u mean by "repeat four times".. repeat what 4 times>?

Let G = (G, . , e), H = (H , * , E) be groups ... (e is the identity) the direct product is defined by: G x H = (GXH, o , (e,E)) where, (g1,h1) o (g2,h2) = (g1 . g2, h1*h2) Question: Show formally that G x H is a group. when it says, "show formally that G x H is a group".. does it mean...

oh just one question though. for V_2(2) V_2x= \left[\begin{array}{ccc}0 & 0 & 1 \\0 & 0 & 0\\0 & 0 & 0\end{array}\right]\left[\begin{array}{c} x \\ y \\ z\end{array}\left]= \left[\begin{array}{c} z \\ 0 \\ 0\end{array}\right]= \left[\begin{array}{c} 0 \\ 0 \\ 0\end{array}\right]...

thnx. this was a brilliant explanation :) really helped me :) brilliant

i don't know the definitions.. i don't know how they got the kernels, the span etc etc i also would appreciate what the difference between generalised eigenspace and normal eigenspace is? thanx

Hi there, I'm having a bit of a problem understanding eigenspaces, kernel and span. I've searched the net and wikipedia but there doesn't seem to be any clear examples. I have an example in a book that says this: Let, A = [ 2 2 2 ] [ 0 2 2 ] [ 0 0 2 ] I can see the characteristic...

How would you prove the following: let V be an inner product space. For v,w \epsilon V we have: |<v,w>| \leq ||v|| ||w|| with equality if and only if v and w are linearly dependent. ----------------------------------------------------------------- So far I know that the...

Suppose f is an entire function such that f(z) = f(z+2\pi) and f(z)=f(z+2\pi i) for all z \epsilon C. How can you use Liouville's theorem to show f is constant.. any help on that please to get me started off.. thnx a lot :)