# Find all subgroups of the given group

• twoflower
In summary, the conversation is about finding all subgroups of a given group, specifically the group (\mathbb{Z}_{12}, +, 0). The person discussing has attempted to find all subgroups by looking at the subgroups generated by each individual element, but wonders if this method may have missed some subgroups. Another person suggests that there is no better method than working them out by hand, and that it is up to the individual to determine if there are non-cyclic subgroups. They also suggest using the fact that the inverse image of a subgroup is a subgroup.
twoflower
Hi all,

I've been practising some algebra excercises and don't know how to solve this one:

Given the group $(\mathbb{Z}_{12}, +, 0)$, find all its subgroups. How many elements will have these subgroups?

The only idea which came to my mind is to see the subgroups generated by each individual element of the given group. So I got:

$$\langle0\rangle = \{0\}$$

$$\langle1\rangle = \mathbb{Z}_{12}$$

$$\langle2\rangle = \{0,2,4,6,8,10\}$$

$$\langle3\rangle = \{0,3,6,9\}$$

$$\langle4\rangle = \{0,4,8\}$$

$$\langle5\rangle = \mathbb{Z}_{12}$$

$$\langle6\rangle = \{0,6\}$$

$$\langle7\rangle = \mathbb{Z}_{12}$$

$$\langle8\rangle = \{0,4,8\}$$

$$\langle9\rangle = \{0,3,6,9\}$$

$$\langle10\rangle = \{0,2,4,6,8,10\}$$

$$\langle11\rangle = \mathbb{Z}_{12}$$

So after removing duplicities, I have 6 subgroups:

$$\mathbb{Z}_{12}$$

$$\{0\}$$

$$\{0,6\}$$

$$\{0,4,8\}$$

$$\{0,3,6,9\}$$

$$\{0,2,4,6,8,10\}$$

But since my approach was rather non-rigorous, I wonder if these are all subgroups of the original subgroup. Is there a chance (when using this approach) that I miss some subgroup?

It seems that I can take care only of elements of the original group, which have some common divisor with module (12) greater than 1, because only they seem to generate some interesting subgroups, other elements (5,7,9,10,11) generate always the whole group. Is this also a rule?

I just would like to know some more general approach to find all subgroups of given group, because if the group gets more complicated than this example, this doesn't seem to be comfortable way to do that.

Best regards.

Last edited:
There really is no better method than simply working them out by hand like this. Sorry if you wanted a clever way of doing it. In general there isn't.

matt grime said:
There really is no better method than simply working them out by hand like this. Sorry if you wanted a clever way of doing it. In general there isn't.

Thank you matt. Is it true that I'll get all subgroups doing it this way? Can't I miss any? And if so, may I restrict myself to generating subgroups from elements which have GCD with module of the group > 1?

You have classified the cyclic subgroups. It is now up to you to try to decide if there are non-cyclic subgroups.

if you know the subgroups of Z, you might look at the surjection from Z to Z/n and use the fact that the inverse image of a subgroup is a subgroup.

## 1. How do you determine the subgroups of a given group?

To find all subgroups of a given group, you can follow a systematic approach known as the subgroup lattice method. This involves listing out all possible combinations of elements within the group and checking if they satisfy the properties of a subgroup.

## 2. What are the properties of a subgroup?

A subgroup must satisfy three properties: closure, identity, and inverse. Closure means that the subgroup must contain all possible products of its elements. Identity means that the subgroup must contain the identity element of the original group. Inverse means that every element in the subgroup must have an inverse element also contained in the subgroup.

## 3. Can a subgroup have a different order than its parent group?

Yes, a subgroup can have a different order than its parent group. In fact, the order of a subgroup must be a divisor of the order of its parent group. This is known as Lagrange's theorem.

## 4. How can I determine if a given subset is a subgroup?

To determine if a given subset is a subgroup, you can check if it satisfies the three properties mentioned earlier: closure, identity, and inverse. If it satisfies all three properties, then it is a subgroup.

## 5. Are there any other methods for finding subgroups?

Yes, there are a few other methods for finding subgroups, such as the factor group method, the normal subgroup method, and the conjugacy class method. These methods may be more efficient for certain types of groups, but the subgroup lattice method is the most commonly used approach.

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