# group theory

• Jan 17th 2007, 06:04 AM
edgar davids
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!!?
• Jan 17th 2007, 06:23 AM
ThePerfectHacker
What does this have to do with linear algebra?
Quote:

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? $\displaystyle n$? (Again finite I think)
Since they are isomorphic they certainly have the same number of elements.
Thus, $\displaystyle |G(n)|=|H(n)|$, then certainly a weaker statement $\displaystyle |G(n)|\equiv |H(n)|(\mbox{ mod }n)$
Quote:

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 $\displaystyle C$? The cyclic group?

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

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