I'm requested to classify the abelian groups $A$ of order $2^5 \times 3^5 $ where :
$| A/A^4 | = 2^4 $
$ |A/A^3 | = 3^4 $
I need to write down the canonical form of each group .
My question is, what does it mean $| A/A^4 |$ ? I understand that the meaning is to: $$G⁄H= \{Hg \mid g\in G\}$$
but the usage of it for classifying the groups is not clear to me.
How does it help me with this case ?
Regards
EDIT:
Suppose that I say $A=B \times C$ where $|B|=2^5$ and $|C|=3^5$ and $|A|=2^5 * 3^5$ .
Now , the goals are :
- $|A|/|A|^3 = (2^5 * 3^5) /|A|^3 = 3^4 $
:meaning I need to make $|A^3|=3*2^5$
- $|A|/|A|^4 = (2^5 * 3^5) /|A|^4 = 2^4 $
:meaning I need to make $|A^4|=2*3^5$
So now after A=BC , how can I find the exact $|A^4|$ and $|A^3|$ ?
thanks again