Exactly n elements of order dividing n in a finite solvable group implies the elements form a subgroup
Suppose is a Finite solvable group (?), and is a natural number dividing the order of . If there are exactly elements of whose power is the identity element, then these elements form a subgroup. This subgroup is clearly a normal subgroup. In fact, it is a characteristic subgroup, and even better, is a fully characteristic subgroup, a homomorph-containing subgroup and a variety-containing subgroup.
- Number of nth roots is a multiple of n
- Number of nth roots of any conjugacy class is a multiple of n
- At most n elements of order dividing n implies every finite subgroup is cyclic
- Frobenius conjecture on nth roots: The conjecture states that the assumption of solvability can be dropped.
The table below lists key facts used directly and explicitly in the proof. Fact numbers as used in the table may be referenced in the proof. This table need not list facts used indirectly, i.e., facts that are used to prove these facts, and it need not list facts used implicitly through assumptions embedded in the choice of terminology and language.
|Fact no.||Statement||Steps in the proof where it is used||Qualitative description of how it is used||What does it rely on?||Difficulty level||Other applications|
|1||Minimal normal implies elementary abelian in finite solvable||Setting up the elementary abelian normal subgroup of , happens before we split into cases.||The proof is inductive, and the goal is to induct from to .||Basic/intermediate group theory||click here|
|2||Lagrange's theorem: We in particular are interested in the version which states that if is a subgroup of , , where is the quotient set. When is a normal subgroup, is the quotient group.||Step (4) of the divides , Step (1) of the other case||We use it to compute the order of the quotient group , from which we ultimately induct. The idea is to try to show that the inductive hypothesis conditions apply to .||Basic group theory||2||click here|
|3||Number of nth roots is a multiple of n (when divides the group order)||Step (6) of case, Step (2) of the other case||Applied to the quotient group to show that the number of roots of a certain kind of is at least a certain amount, which is then played off against it being at most a certain amount.||Basic/intermediate group theory||click here|
|4||Solvability is quotient-closed||Step (13) of case, Step (6) of other case||Show conditions prevail to apply inductive hypothesis to quotient group||Basic group theory||?Difficulty level||click here|
|5||Solvability is subgroup-closed||Step (8) of does not divide case||Show solvability of a subgroup of||Basic group theory||click here|
|6||Hall subgroups exist in finite solvable||Step (8) of does not divide case||Find a subgroup of order in a subgroup where the -part is||Advanced||click here|
|7||Order of element divides order of group||Step (9) of does not divide case||After finding subgroup of order , show it is precisely the subgroup we want||Basic group theory|| click here
This proof uses the principle of mathematical induction in a nontrivial way (i.e., it would be hard to write the proof clearly without explicitly using induction).
We do the proof by induction. Specifically, we assume that the statement holds true for all finite solvable groups of strictly smaller orders than .
PROOF OF INDUCTIVE STEP:
Given: A finite solvable group of order . A natural number . is the set of roots of unity in (i.e., the set of elements of order dividing , and has elements.
To prove: is a subgroup.
Proof: By fact (1), has a nontrivial elementary abelian normal subgroup of order for some prime power dividing the order of . We split the proof into two cases, based on whether divides or not.
Case that divides
Case that does not divide
Proof that the subgroup is normal, characteristic and fully invariant
This follows from the fact that if is a homomorphism from to any subgroup of , then . Hence, if , so is .