Quotient Groups and their Index

In summary, the conversation discusses solving a difficult exercise involving quotient groups and proving the equation |G:H|=|G:K||K:H|. It also mentions the use of Lagrange's theorem and the concept of lattice isomorphism. The conversation ends by discussing the importance of counting the number of cosets in solving the exercise and considering both directions of the equation.
  • #1
Pjennings
17
0
As a way to keep busy in between semesters I decided to work my way through Algebra by Dummit and Foote in order to prepare for the fall. Working my way through quotient groups is proving to be quite difficult and as a result I'm stuck on an exercise that looks simple, but I just don't know where to start. Any ideas how to prove that given H[tex]\leq[/tex]K[tex]\leq[/tex]G |G:H|=|G:K||K:H|?
 
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  • #2
Have they not gone through Lagrange's theorem yet?
 
  • #3
eok20 said:
Have they not gone through Lagrange's theorem yet?
Who said any of the groups were finite?
 
  • #4
Martin Rattigan said:
Who said any of the groups were finite?

good point.
 
  • #5
Lattice isomorphism theorem. Assuming those are normal subgroups, (G/K)/(K/H) is isomorphic to G/H.

Of course, it's not hard to just count the number of cosets. G = g_1K + ... + g_nK where n = [G:K]. K = k_1H + ... + k_mH, where m = [H:K]. Then G = g_1(k_1 + ... + k_m)H + ... + g_n(k_1+...+k_m)H, where I have abused notation a little. This shows [G:H]<=[G:K][K:H]. Now think about the other direction.
 

1. What is a quotient group?

A quotient group is a mathematical concept that involves dividing a group into smaller subgroups based on certain properties or relations. It is denoted as G/N, where G is the original group and N is the subgroup.

2. How is the index of a quotient group determined?

The index of a quotient group is determined by the number of cosets of the subgroup N in the original group G. It is denoted as [G:N].

3. What is the significance of quotient groups and their index?

Quotient groups and their index are important in abstract algebra as they help in understanding the structure and properties of groups. They also have applications in fields such as number theory and geometry.

4. Can the index of a quotient group be greater than the order of the original group?

Yes, the index of a quotient group can be greater than the order of the original group. This happens when the subgroup N does not have a direct relationship with the elements in the group G.

5. How are quotient groups and their index used in real-world problems?

Quotient groups and their index are used in various real-world problems, such as cryptography, coding theory, and data analysis. They can also be applied in studying symmetry in different systems, such as crystals and molecules.

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