This is Exercise 8.28 of Gallian’s “Contemporary Abstract Algebra”.
Answers that use only methods from the textbook prior to the exercise are preferred.
Here $ G_1\oplus G_2$ is the internal direct product of $ G_1$ by $ G_2$ .
Here $ G_1\le G_2$ means $ G_1$ is a subgroup or equal to the group $ G_2$ .
Find a subgroup of $ \Bbb Z_4\oplus\Bbb Z_2$ not of the form $ H\oplus K$ for some $ H\le \Bbb Z_4, K\le \Bbb Z_2$ .
I must confess: I cheated here a little bit by looking up the subgroups of $ \Bbb Z_4\times \Bbb Z_2$ . But notice the difference in notation! I think in terms of just plain old direct products (because aren’t external and internal direct products equivalent? Yes! But this is not established in the textbook yet; indeed, the former is not even mentioned at this point).
It appears to me to be a trick question. Here are the subgroups of $ \Bbb Z_4\times \Bbb Z_2$ . Where is the subgroup of the desired form?
My guess is that there’s some technical aspect of internal direct products that is being emphasised here.
Please help 🙂