On group theory 2
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On group theory 2
Find all the non-isomorphic abelian groups of order \(\displaystyle{300}\) .
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Re: On group theory 2
This is a standard exercise on the structure theorem of finite abelian groups.
Since $300 = 2^2 \cdot 3 \cdot 5^2$ then there are $p(2)^2p(1) = 4$ abelian groups of $300$. Here $p(n)$ denotes the number of partitions of $n$. The groups are (isomorphic to)
$C_{2} \times C_{2} \times C_3 \times C_5 \times C_5$
$C_{2} \times C_{2} \times C_3 \times C_{25}$
$C_{4} \times C_3 \times C_5 \times C_5$
$C_{4} \times C_3 \times C_{25}$
in prime power form.
Since $300 = 2^2 \cdot 3 \cdot 5^2$ then there are $p(2)^2p(1) = 4$ abelian groups of $300$. Here $p(n)$ denotes the number of partitions of $n$. The groups are (isomorphic to)
$C_{2} \times C_{2} \times C_3 \times C_5 \times C_5$
$C_{2} \times C_{2} \times C_3 \times C_{25}$
$C_{4} \times C_3 \times C_5 \times C_5$
$C_{4} \times C_3 \times C_{25}$
in prime power form.
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- Community Team
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Re: On group theory 2
Thank you mr. Demetres.
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