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## Main Question or Discussion Point

Hi, I am trying to find all groups G of order 16 so that for every y in G, we have y+y+y+y=0.

My thought is using the structure theorem for finitely-generated PIDs. So I can find 3:

## \mathbb Z_4 \times \mathbb Z_4##,

## \mathbb Z_4 \times \mathbb Z_2 \times \mathbb Z_2 ## , and:

## \mathbb Z_2 ^4 ## .

How can I tell if these are the only 3 groups with this property up to isomorphism?

Thanks.

My thought is using the structure theorem for finitely-generated PIDs. So I can find 3:

## \mathbb Z_4 \times \mathbb Z_4##,

## \mathbb Z_4 \times \mathbb Z_2 \times \mathbb Z_2 ## , and:

## \mathbb Z_2 ^4 ## .

How can I tell if these are the only 3 groups with this property up to isomorphism?

Thanks.