# Abelian Group/Subgroups of Power Algebra

## Homework Statement

Let X be a set with exact three elements. Then its power algebra P(x), is an Abelian group with symmetrical difference operator delta.
A subset H of P(x) is a subgroup if, for all A, B in H, A deltaB in H.
Find all possible subgroups of P(x).

## The Attempt at a Solution

I have no clue how to begin solving this problem. But I know that the power set of any set X becomes an Abelian group if we use the symmetric difference as operation. How can I show this?

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matt grime
Homework Helper
The power set has only 8 elements. So you can just do it by exhaustion (and a little common sense: subgroups of order 2 are trivial to find, so that just leaves order 4, and there are only two groups of order 4 up to isomorphism).

The power set has only 8 elements. So you can just do it by exhaustion (and a little common sense: subgroups of order 2 are trivial to find, so that just leaves order 4, and there are only two groups of order 4 up to isomorphism).

matt grime
Homework Helper
Start with subgroups of order 2. What is a (sub)group of order 2? It is a set {e,g} with e the identity (which is what in this case?) and g satisfying what?

Just to check where the problem lies: do you understand why the only subgroups have orders 1,2,4, and 8?

Actually with this problem... I am not even sure where to start... I dont know what the subgroup of order 2 is. I have nothing specified apart from the problem statement, which I agree is rather confusing. I do not understand why the only subgroups have the orders 1, 2, 4 and 8?

I read that the identity of the subgroup is the identity of the group. But with this proof... I am totally lost of where I should start. Could you please guide me?

matt grime