A question about linear algebra

[Artusartos]

Classify the abelian groups of order 32.
a) In each case give the annihilator of the group along with $$dim_{Z_2} \frac{ker \mu_{2^s}}{ker \mu_{2^{s-1}}}$$


for s=1,...,5. Where $$\mu_k(x) = kx$$ for all k.

b) If you know the annihilator of each of these groups, how many values of s (beginning with s=1) are needed to tell them apart?

My answer:

The abelian groups of order 32:

$$Z_{32}$$

$$Z_{16} \bigoplus Z_2$$

$$Z_8 \bigoplus Z_4$$

$$Z_8 \bigoplus Z_2 \bigoplus Z_2$$

$$Z_4 \bigoplus Z_2 \bigoplus Z_2 \bigoplus Z_2$$

$$Z_2 \bigoplus Z_2 \bigoplus Z_2 \bigoplus Z_2 \bigoplus Z_2$$


For part a, if we look at $$Z_{32}$$, we have

$$dim_{Z_2} \frac{ker \mu_{2^s}}{ker \mu_{2^{s-1}}}$$

=$$dim_{Z_2} \frac{Z_\bar{16}}{Z_\bar{32}}$$

I'm kind of stuck now...can anybody please give me a hint?

Thanks in advance

 

[lippyka]

!For part a, we haveZ_{32}: Annihilator = 0, dim_{Z_2} \frac{ker \mu_{2^s}}{ker \mu_{2^{s-1}}} = 0Z_{16} \bigoplus Z_2: Annihilator = 2, dim_{Z_2}\frac{ker \mu_{2^s}}{ker \mu_{2^{s-1}}} = 0Z_8 \bigoplus Z_4: Annihilator = 4, dim_{Z_2}\frac{ker \mu_{2^s}}{ker \mu_{2^{s-1}}} = 1Z_8 \bigoplus Z_2 \bigoplus Z_2: Annihilator = 8, dim_{Z_2}\frac{ker \mu_{2^s}}{ker \mu_{2^{s-1}}} = 2Z_4 \bigoplus Z_2 \bigoplus Z_2 \bigoplus Z_2: Annihilator = 16, dim_{Z_2}\frac{ker \mu_{2^s}}{ker \mu_{2^{s-1}}} = 3Z_2 \bigoplus Z_2 \bigoplus Z_2 \bigoplus Z_2 \bigoplus Z_2: Annihilator = 32, dim_{Z_2}\frac{ker \mu_{2^s}}{ker \mu_{2^{s-1}}} = 4For part b, we need at least 5 values of s to tell the groups apart (s=1,2,3,4,5).