Find Subgroups of Z3xZ3: Solutions and Equations for Homework"

[FanofAFan]

Homework Statement

Find all the subgroups of Z₃xZ₃

Homework Equations

The Attempt at a Solution

So H is a subgroup of G if H is nonempty, closure for a group, and the inverse must be in H.

(identity)
{([0],[0]) ([0],[1]) ([0],[2])}
{([0],[0]) ([1],[0]) ([2],[0])}
{([0],[0]) ([1],[1]) ([2],[2])}
{([0],[0]) ([0],[1]) ([0],[2]) ([1],[0]) ([2],[0]) ([1],[1]) ([2],[2]) ([1],[2]) ([2],[1])}
{([0],[0]) ([1],[2]) ([2],[1])

is there more subgroups or do I have them all

 

[mighty2000]

?

Hi there, great question! It looks like you have found all the subgroups of Z3xZ3, but let me explain a bit more about how to determine subgroups in general.

As you mentioned, a subgroup H of a group G must satisfy three conditions: it must be nonempty, closed under the group operation, and contain the inverse of each of its elements.

To determine all subgroups of a given group, we can start by looking at the identity element of the group. In this case, the identity element of Z3xZ3 is ([0],[0]). This means that every subgroup must contain this element.

Next, we can look at the elements that have inverses in Z3xZ3. These are ([0],[1]) and ([0],[2]). This means that every subgroup must also contain these elements and their inverses ([0],[2]) and ([0],[1]), respectively.

Now, we can start building subgroups by adding in other elements of Z3xZ3 and their inverses, making sure to always include the identity element and the inverses of the elements we add in.

Using this method, you have correctly found all the subgroups of Z3xZ3. Great job!