Difference between revisions of "Index is multiplicative"
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+ | {{subgroup metaproperty satisfaction| | ||
+ | property = subgroup of finite index| | ||
+ | metaproperty = transitive subgroup property}} | ||
+ | |||
==Statement== | ==Statement== | ||
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Suppose <math>K \le H \le G</math> are groups such that the [[fact about::index of a subgroup|indices]] <math>[G:H]</math> and <math>[H:K]</math> are finite. Then, we have: | Suppose <math>K \le H \le G</math> are groups such that the [[fact about::index of a subgroup|indices]] <math>[G:H]</math> and <math>[H:K]</math> are finite. Then, we have: | ||
+ | |||
+ | <math>[G:K] = [G:H][H:K]</math>. | ||
+ | |||
+ | In particular, if <math>K</math> has finite index in <math>H</math> and <math>H</math> has finite index in <math>G</math>, then <math>K</math> has finite index in <math>G</math>. | ||
+ | |||
+ | Note also that if <math>[G:K]</math> is finite, then so are the other two indices, and we thus get: | ||
<math>[G:K] = [G:H][H:K]</math>. | <math>[G:K] = [G:H][H:K]</math>. |
Revision as of 13:23, 1 November 2008
This article gives the statement, and possibly proof, of a subgroup property (i.e., subgroup of finite index) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about subgroup of finite index |Get facts that use property satisfaction of subgroup of finite index | Get facts that use property satisfaction of subgroup of finite index|Get more facts about transitive subgroup property
Contents
Statement
Set-theoretic version
Suppose are groups. Then, we have a natural surjective map between the left coset spaces:
with the property that the inverse image of each point has size equals to the size of the left coset space .
Numerical version
Suppose are groups such that the indices and are finite. Then, we have:
.
In particular, if has finite index in and has finite index in , then has finite index in .
Note also that if is finite, then so are the other two indices, and we thus get:
.
Note that the statement makes sense even for infinite groups, if we interpret the cardinalities of the coset spaces as infinite cardinals and do the multiplication accordingly.
Related facts
- Lagrange's theorem: Lagrange's theorem is a special case of this where is the trivial group.
- Third isomorphism theorem: The third isomorphism theorem is a stronger version of the statement where both subgroups are normal in the whole group. In this case, the surjective map from to is a homomorphism and the kernel is .
Facts used
- Subgroup containment implies coset containment: If ,then every left coset of is contained in a unique left coset of .
Proof
Proof of the set-theoretic version
Given: Groups .
To prove: There is a surjective map from to where the inverse image of every point has size equal to the size of .
Proof: Define as the map sending a coset of to the unique coset of containing it (fact (1)). In other words:
.
is a well-defined map from to . Further:
- is surjective, since for any coset of in , .
- The size of each inverse image equals the size of : Consider a coset . We want to find all the left cosets of which map to this. This is equivalent to finding all the left cosets of contained in .
For this, consider a map that sends a left coset of in , to the left coset .
Note that:
- This map is well-defined, because if is replaced by , the left cosets and are the same.
- The map sends left cosets of in to left cosets of in , because since , .
- The map is injective, because it comes from a left multiplication on cosets.
- The map is surjective, because any left coset of in arises as the image of the left coset , which lies in .
Thus, is a bijection between the left cosets of in and the left cosets of in . Thus, the number of left cosets of in equals the size of the left coset space .
Proof of the numerical version
The set-theoretic version shows that is the disjoint union of sets, each of size . This yields:
.
By the definition of index of a subgroup, this yields:
.
References
Textbook references
- Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 77, Exercise 4 of Miscellaneous Problems (asked only for a finite group)