# Abelian Group/Subgroups of Power Algebra

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In summary: When you have a set with 1 element, the identity element is the set itself. So the Cayley table for the group is just {1}. When you have a set with 2 elements, the identity element is the set {1,1}. And so on.The first step is to identify the identity element of the group. The identity element of a group is the set itself.

## 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?

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 said:
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).

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 don't 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?

You should start with the basics of group theory: the definition of a group, subgroup, Lagrange's theorem, for example. If you don't know what a subgroup of order 2 is, then you'll never find one.

I don't think there is anything confusing about the question if you know what the definitions are.

First, identify the identity element in the group. Actually, first, what are the elements of the group? What is the Cayley table (or multiplication table) for the group? If a set with 3 elements is too big, try a set with 1 element, then 2 elements. Just take two elements in the group and compose them, see what happens.

## What is an Abelian Group?

An Abelian group, also known as a commutative group, is a mathematical structure consisting of a set of elements and an operation that satisfies the commutative property. This means that the order in which the operation is performed on two elements does not affect the result. The group must also satisfy the properties of associativity, identity, and inverse.

## What is a Subgroup?

A subgroup is a subset of an Abelian group that also satisfies the properties of a group. This means that the subset must also have an operation that is associative, has an identity element, and each element must have an inverse. Additionally, the subset must be closed under the operation of the original group.

## How do you determine if a subset is a Subgroup of an Abelian group?

To determine if a subset is a subgroup of an Abelian group, you must check if it satisfies the properties of a group. This includes checking for closure, associativity, identity, and inverse. If all of these properties are satisfied, then the subset is considered a subgroup of the original group.

## What is the Power Algebra of an Abelian Group?

The Power Algebra of an Abelian group is the set of all elements that can be expressed as a power of a given element in the group. This means that the set includes all possible combinations of the given element raised to different powers. This algebraic structure is useful in studying the structure and properties of Abelian groups.

## How are Abelian Group/Subgroups of Power Algebra used in real-world applications?

Abelian groups and their subgroups, as well as power algebra, have various applications in different fields such as physics, chemistry, and computer science. In physics, they are used to study and describe symmetries in physical systems. In chemistry, they are used to classify and predict the properties of molecules. In computer science, they are used in cryptography and error-correcting codes.