calculus of subgroup orders
Recall that for any group and any subgroup of we can define left and/or right cosets of in . As the cardinality of left cosets equals the cardinality of right cosets, the index, denoted is well-defined as this cardinality.
When is a finite group then we can rewrite this in the familiar form (as actually proved by LaGrange)
and so conclude the familiar statement: “The order of every subgroup divides the order of the group.”
The first corollary to the theorem states that given then
So when is finite we have
This should be contrasted with the third isomorphism theorem which claims if and are normal in then
It is preferrable to express the various equations relating indices of subgroups with multiplications. This is to allow for infinite indices, as we can multiply cardinal numbers, but we may not always be able to make sense of cardinal number division.
When only finite groups are considered so that division is allowed, expressing the theorems as quotients is often easier to understand.
Next suppose and are any two subgroups of then we define
(sometimes called the complex of and .) Caution: it is not always true that is a subgroup of . It is true if either or is a normal subgroup of and occassionally it is true even without or being normal – for example when so called permutable subgroups.
If and are finite subgroups then we can express this as:
Once again if we have normality, say is normal in , then this is mimicks second isomorphism theorem:
even when all the groups are infinite. Furthermore, if is a subgroup of then we can write
Note the diagram is a Hasse diagram of the lattice of subgroups of . We further inforce a policy of drawing edges of the same length if the index of the corresponding subgroups are equal. Thus (9) is simply a proof that the picture is accurate: opposite sides of a parallelogram are congruent.
The most common use of index calculus is for subgroups of finite index in . This allows one to solve for indices from given assumptions. It is also quite common to prove certain configurations of subgroups are impossible as the indices are relatively prime.
|Title||calculus of subgroup orders|
|Date of creation||2013-03-22 15:48:12|
|Last modified on||2013-03-22 15:48:12|
|Last modified by||Algeboy (12884)|
|Synonym||Theorem of Lagrange|
|Defines||complex of subgroups|