I'm preparing for a test on Monday.
This is an exercise in the past homework.
Find all subgroups of $\mathbb{Z}_2\times \mathbb{Z}_2\times\mathbb{Z}_4$ isomorphic to $Z_2\times Z_2$.
Is there a way to solve this effectively?
2026-04-04 06:37:59.1775284679
How do I effectively solve this kinds of problems?
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It's pretty clear that every pair of distinct elements with order 2 will generate such a subgroup. Seek all of those pairs!
If I didn't miscount, I think I see 7 elements of order 2. The subgroup will contain three elements of order 2. Experiment and see what combinations are distinct.
If you try all $_7C_2$ pairs, you should find that everything is repeated on the list three times. That's because the group the pair generates only has one other element of order 2: the sum of the generators. So, you can expect 7 distinct subgroups in your final result.
Be warned though: it won't be this simple for higher orders and different isotypes. This works well here because the order is so small.