What is a Quotient Group? A Simple Explanation

In summary, a quotient group is a group formed by partitioning a larger group and then assigning a group structure to the resulting subsets or cosets. This is done using an operation, such as addition, on the representatives of each subset. In the example of the numbers on a clock, the integers are partitioned into 12 sets, or cosets, and the group structure is inherited from the integers through addition. This allows for a simpler representation of the larger group.
  • #1
Solid Snake
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Can someone please explain to me, in as simple words as possible, what a quotient group is? I hate my books explanation, and I would love it if someone can tell me what it is in english?
 
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  • #2
Do you know what "equivalence relations" and "equivalence classes" are?
 
  • #3
Solid Snake said:
Can someone please explain to me, in as simple words as possible, what a quotient group is? I hate my books explanation, and I would love it if someone can tell me what it is in english?

My favorite quotient groups is the numbers on a face clock.

The numbers ...-11,1,13,25... are an equivalence class, with representative 1.

Thus, the infinite set of integers ...-1,0,1,2,3,... is partitioned into 12 sets, or cosets. Since the theorem of quotient groups holds, these 12 representatives have a group structure inherited from the group structure of the integers. The operation in question is addition. So, let's take two representatives in the quotient group, say 8 and 7, then 8+7 is 15, however we might prefer to represent 15 with another integer in it's class, say 3. This is how we define the inherited operation of addition in the quotient group.

Let's see how this corresponds with the steps used in general quotient groups. What steps do we use to get from the integers to the group with elements {1,2,...,12}, in other words, how do we construct the latter group. Let Z represent the group of integers, let 12Z represent the subgroup in Z generated by 12, ie 12Z={...-12,0,12,24,36,...}. Then the numbers on the clock are constructed via the symbolism

Z/(12Z).

Oops, got to go, I'll let others continue this. Please let us know where you're at on this.
 

1. What is a quotient group?

A quotient group is a mathematical concept that represents a group formed by partitioning a larger group into smaller subgroups. It is typically denoted by G/H, where G is the original group and H is the subgroup being used for partitioning.

2. How is a quotient group different from a regular group?

A quotient group is different from a regular group because it is formed by dividing a larger group into smaller subgroups, whereas a regular group is not divided or partitioned in any way.

3. What is the purpose of a quotient group?

The purpose of a quotient group is to simplify the study of a larger group by breaking it down into smaller, more manageable subgroups. This allows for easier analysis and understanding of the group's properties and characteristics.

4. How is a quotient group related to cosets?

A quotient group is closely related to cosets, as cosets are the individual subgroups that are formed when a larger group is partitioned. The elements of a quotient group are the cosets of the original group.

5. Can you give an example of a quotient group?

Yes, an example of a quotient group would be the integers (Z) divided by the even integers (2Z). The quotient group, denoted as Z/2Z, would consist of two cosets: one containing all the even integers and one containing all the odd integers. These two cosets together make up the quotient group Z/2Z.

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