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I have two questions:

1. How to list(up to isomorphism) all finite abelian groups of order 675, which don't have elements of order 45?

2. How to list(up to isomorphism) all finite abelian groups of order 196, which have cyclic group of order 98?

I just don't understand how to start. Please help me understand what's going on.

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  • $\begingroup$ Factor the order of the group. As user627482 suggests, the fundamental theorem will give you all nonisomorphic abelian groups of that order (be mindful that $ \mathbb{Z}_p \times \mathbb{Z}_q \cong \mathbb{Z}_{pq}$if $p$ and $q$ are relatively prime). Now decide which of these possibilities satisfy the criteria you are interested in. $\endgroup$ – Chris Leary Jun 8 at 23:13
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Use the fundamental theorem of finite abelian groups

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