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Magazine Problem 7
$$ \operatorname{In} \mathbb{Z}_{2} \times \mathbb{Z}_{4} \text { find a subgroup } H \text { such that } H \approx \mathbb{Z}_{2} \times \mathbb{Z}_{2} \text { . } $$
Problem 7
In Exercises 7 through 10 find the distinct cosets of the indicated subgroup in the indicated group. \(H=\langle(8,2)\rangle\) in \(\mathbb{Z}_{10} \times \mathbb{Z}_{4}\)
Problem 7
Find up to isomorphism all Abelian groups of the indicated orders. $$ n=60 $$
Problem 8
Find up to isomorphism all Abelian groups of the indicated orders. $$ n=60 $$
Problem 8
$$ \text { In } D_{4} \text { find a subgroup } H \text { such that } H \approx \mathbb{Z}_{2} \times \mathbb{Z}_{2} \text { . } $$
Problem 9
Find up to isomorphism all Abelian groups of the indicated orders. $$ n=72 $$
Problem 9
In Exercises 7 through 10 find the distinct cosets of the indicated subgroup in the indicated group. \(H=\langle(6,8)\rangle\) in \(3 Z \times 2 Z\)
Problem 9
Explain why there are no nontrivial proper subgroups \(H_{1}, H_{2}\), and \(H_{3}\) in \(\mathbb{Z}_{36}\) such that \(\mathrm{Z}_{36}=\mathrm{H}_{1} \oplus \mathrm{H}_{2} \oplus \mathrm{H}_{3}\).
Problem 9
$$ \begin{aligned} &\text { In } \mathbb{Z}_{4} \times \mathbb{Z}_{4} \text { find two subgroups } H \text { and } K \text { of order } 4 \text { such that } H \text { is not isomorphic }\\\ &\text { to } K \text { , but }\left(\mathbb{Z}_{4} \times \mathbb{Z}_{4}\right) / H \cong\left(\mathbb{Z}_{4} \times \mathbb{Z}_{4}\right) / K \text { . } \end{aligned} $$
Problem 10
Find up to isomorphism all Abelian groups of the indicated orders. $$ n=108 $$
