Find All Subgroups of U(14) in $\mathbb{Z}_{14}$

  • MHB
  • Thread starter alexmahone
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In summary, there are four subgroups of U(14), the group of units of $\mathbb{Z}_{14}$. These subgroups have orders of 1, 2, 3, and 6, and are generated by a single element. The group is abelian and cyclic, with only two possible groups of order 6, one of which is abelian.
  • #1
alexmahone
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Find all subgroups of U(14), the group of units of $\mathbb{Z}_{14}$.

My attempt:

U(14)={1, 3, 5, 9, 11, 13} and the operation is multiplication modulo 14. Do I just find the subgroups generated by each of the elements in U(14)? What about subgroups generated by two elements?
 
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  • #2
You have an abelian group of order $6$. There are, up to isomorphism, only two groups of order $6$, and only one of those is abelian. Thus, your group is cyclic.

This means you have exactly:

1 group of order 1
1 group of order 2
1 group of order 3
1 group of order 6

each of which will be generated by a single element. I suggest finding a generator (element) of order 6, first.
 

Related to Find All Subgroups of U(14) in $\mathbb{Z}_{14}$

1. What is the definition of a subgroup?

A subgroup is a subset of a group that satisfies all the axioms of a group, including closure, associativity, identity element, and inverse elements.

2. How do you find all subgroups of a given group?

To find all subgroups of a given group, you can start by identifying the elements that are in the subgroup and then checking if they satisfy the group axioms. Alternatively, you can use the subgroup criterion, which states that if a subset of a group contains the identity element and is closed under the group operation, then it is a subgroup.

3. What is the group operation in $\mathbb{Z}_{14}$?

The group operation in $\mathbb{Z}_{14}$ is addition modulo 14. This means that you add two elements and then take the remainder when divided by 14.

4. How many subgroups are there in $\mathbb{Z}_{14}$?

There are 6 subgroups in $\mathbb{Z}_{14}$. These include the trivial subgroup (containing only the identity element), the entire group, and four other subgroups of order 2, 7, 2, and 1.

5. What are the elements of the subgroups of U(14) in $\mathbb{Z}_{14}$?

The elements of the subgroups of U(14) in $\mathbb{Z}_{14}$ are the integers that are relatively prime to 14. These include 1, 3, 5, 9, 11, and 13.

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