How to Find the Subgroup of a Quotient Group with a Given Generator?

In summary, a generator of quotient group is an element that can generate all other elements in the quotient group by repeated multiplication or combination. It is different from a generator of a group, as it generates elements in the quotient group rather than the original group. An element is considered a generator of quotient group if its powers can generate all other elements in the quotient group and if its order is equal to the order of the quotient group. A quotient group can have multiple generators, meaning there are multiple elements that can generate all other elements. The concept of generator of quotient group is important because it helps us understand the structure and properties of quotient groups, simplifies calculations and proofs, and allows us to focus on the generators instead of all the elements in the
  • #1
physix123
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i am confused about how to find the subgroup of a quotient group given a generator. for example, a lot of problems give as the group Z/nZ with n very large. how do you find the subgroup given a generator?

thanks!
 
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  • #2
Given a generator of the subgroup? Just enumerate the elements you obtain by performing the group's operation on the generator until you get the identity...or am I misunderstanding what you are asking...?
 

1. What is a generator of quotient group?

A generator of quotient group is an element that can generate all other elements in the quotient group by repeated multiplication or combination.

2. How is a generator of quotient group different from a generator of a group?

A generator of quotient group is an element in the quotient group, while a generator of a group is an element in the original group. A generator of quotient group generates elements in the quotient group, while a generator of a group generates elements in the original group.

3. How do you determine if an element is a generator of quotient group?

An element is a generator of quotient group if its powers can generate all other elements in the quotient group and if its order is equal to the order of the quotient group.

4. Can a quotient group have multiple generators?

Yes, a quotient group can have multiple generators. This means that there are multiple elements in the quotient group that can generate all other elements.

5. Why is the concept of generator of quotient group important?

The concept of generator of quotient group is important because it helps us understand the structure and properties of quotient groups. It also allows us to simplify calculations and proofs in quotient groups by focusing on the generators instead of all the elements in the group.

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