1. ## group theory

can someone help me with this questuon please its fundamnetal to the topic and preventing me understand the topic!!

Thanks

Edgar

if G is a group and n > or equal to 1 define

G(n) = {x element of G : ord (x) = n}

If G is isomorphic to H show that for all n greater than or equal to 1
mod G(n) = mod H(n)

Deduce that C(subscript 3) * C(subscript 3) is not isomorphic to C(subscript 9)

Is it tru that C 3 * C 5 is isomorphic to C15

whats going on here!!?

2. What does this have to do with linear algebra?
Originally Posted by edgar davids
can someone help me with this questuon please its fundamnetal to the topic and preventing me understand the topic!!

Thanks

Edgar

if G is a group and n > or equal to 1 define

G(n) = {x element of G : ord (x) = n}

If G is isomorphic to H show that for all n greater than or equal to 1
mod G(n) = mod H(n)
Mod, what? $n$? (Again finite I think)
Since they are isomorphic they certainly have the same number of elements.
Thus, $|G(n)|=|H(n)|$, then certainly a weaker statement $|G(n)|\equiv |H(n)|(\mbox{ mod }n)$
Deduce that C(subscript 3) * C(subscript 3) is not isomorphic to C(subscript 9)

Is it tru that C 3 * C 5 is isomorphic to C15

whats going on here!!?
What is $C$? The cyclic group?

You can just use the fundamental theorem,
$C_3 \times C_3 \not \simeq C_9$
Because they are distinct prime power decompositions.

$C_3\times C_5 \simeq C_{15}$
Because $\gcd(3,5)=1$.