Classify the abelian groups of order 32.(adsbygoogle = window.adsbygoogle || []).push({});

a) In each case give the annihilator of the group along with [tex]dim_{Z_2} \frac{ker \mu_{2^s}}{ker \mu_{2^{s-1}}}[/tex]

for s=1,...,5. Where [tex]\mu_k(x) = kx [/tex] 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:

[tex]Z_{32}[/tex]

[tex] Z_{16} \bigoplus Z_2 [/tex]

[tex] Z_8 \bigoplus Z_4[/tex]

[tex] Z_8 \bigoplus Z_2 \bigoplus Z_2 [/tex]

[tex] Z_4 \bigoplus Z_2 \bigoplus Z_2 \bigoplus Z_2 [/tex]

[tex] Z_2 \bigoplus Z_2 \bigoplus Z_2 \bigoplus Z_2 \bigoplus Z_2[/tex]

For part a, if we look at [tex]Z_{32}[/tex], we have

[tex]dim_{Z_2} \frac{ker \mu_{2^s}}{ker \mu_{2^{s-1}}}[/tex]

=[tex]dim_{Z_2} \frac{Z_\bar{16}}{Z_\bar{32}}[/tex]

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

Thanks in advance

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# Homework Help: A question about linear algebra

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