Generating Subgroups in <Z\stackrel{X}{13}> Modulo 13 Under Multiplication

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In summary, a subgroup is a smaller group nested within a larger group that satisfies the four group axioms. It can be generated by a subset of elements from the larger group, known as the generating set. A subgroup can be generated by multiple groups, and it is the smallest subgroup contained in all of the generating groups. The significance of subgroups generated by groups lies in their ability to help us understand the structure and properties of a larger group and study different aspects of it. These subgroups can also be cyclic, generated by a single element, or by multiple elements. Every subgroup of a cyclic group is also a cyclic group.
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hitmeoff
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Homework Statement


In the group <Z[tex]\stackrel{X}{13}[/tex]> of nonzero classes modulo 13 under multiplication, find the subgroup generated by [tex]\overline{3}[/tex] and [tex]\overline{10}[/tex]

Homework Equations


The Attempt at a Solution


Doesnt 3 generate {3,6,9,12} and 10 generate {2,5,10}?
 
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  • #2
The problem says says "under multiplication" so the subgroup generated by [itex]\overline{3}[/itex] includes all products of [itex]\overline{3}[/itex]. You are adding: 3+ 3= 6, etc. 3*3= 9 and 3*9= 27= 1 (mod 13)
 
  • #3
3*3 = 9 (13)
3*3*3 =27 = 1 (13)
3*3*3*3 = 81 = 3(13)
3*3*3*3*3 = 243 = 9(13)
3*3*3*3*3*3 = 1 (13)

Do this, and then check that the results are allowed given the constraints you can infer from the order of the Cyclic Group.
 

1. What is a subgroup?

A subgroup is a subset of a group that satisfies the four group axioms: closure, associativity, identity element, and inverse element. It is a smaller group nested within a larger group.

2. How is a subgroup generated by a group?

A subgroup is generated by a group when a subset of the group's elements can be used to create a smaller group that satisfies the four group axioms. This subset is called the generating set.

3. Can a subgroup be generated by more than one group?

Yes, a subgroup can be generated by more than one group. This is known as a joint subgroup or an intersection of subgroups. It is the smallest subgroup that is contained in all of the generating groups.

4. What is the significance of subgroups generated by groups?

Subgroups generated by groups are important because they help us understand the structure and properties of a larger group. They also allow us to study different aspects of the group by looking at its subgroups.

5. How are subgroups generated by groups related to cyclic groups?

Cyclic groups are a type of group that can be generated by a single element. Subgroups generated by groups can also be cyclic, but they can also be generated by multiple elements. In fact, every subgroup of a cyclic group is also a cyclic group.

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