Zero-or-scalar lemma
Contents
Statement
Over the complex numbers
Let be a finite group and
an Irreducible linear representation (?) of
over
. Let
, such that the size of the conjugacy class of
is relatively prime to the degree of
. Then, either
is a scalar or
.
Over a splitting field of characteristic zero
The proof as presented here works only over the complex numbers, but it can be generalized to any splitting field for that has characteristic zero.
Applications
- Conjugacy class of prime power size implies not simple
- Order has only two prime factors implies solvable, also called Burnside's
-theorem (proved via conjugacy class of prime power size implies not simple)
Facts used
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 | Size-degree-weighted characters are algebraic integers: For an irreducible linear representation over ![]() |
Step (1) (in turn used in Step (4), leading to Step (5)) | Helps in showing that ![]() |
Algebraic number theory + linear representation theory | click here | |
2 | Characters are algebraic integers: The character of a linear representation is an algebraic integer. | Step (4), leading to Step (5) | Helps in showing that ![]() |
Basic linear representation theory | 2 | click here |
3 | Element of finite order is semisimple and eigenvalues are roots of unity | Step (6), which in turn is critical to later steps | Critical to understanding ![]() ![]() |
Basic linear representation theory | click here |
Proof
This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format
Given: A finite group , a nontrivial irreducible linear representation
of
over
with character
. An element
with conjugacy class
. The degree of
and the size of
are relatively prime.
To prove: Either or
is a scalar.
Proof: Note that when the symbol appears as an input to a representation or a character, it refers to the identity element of
. When it appears as the output of a character, or in another context, it refers to the real number
.
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | The number ![]() |
Fact (1) | ![]() ![]() ![]() ![]() ![]() |
Given+Fact direct | |
2 | There exist integers ![]() ![]() ![]() |
![]() ![]() ![]() |
By definition of relatively prime. | ||
3 | We get ![]() |
Step (2) | Multiply both sides of Step (2) by ![]() | ||
4 | The expression ![]() |
Fact (2) | Step (1) | [SHOW MORE] | |
5 | ![]() |
Steps (3), (4) | [SHOW MORE] | ||
6 | ![]() ![]() ![]() |
Fact (3) | ![]() |
[SHOW MORE] | |
7 | Every algebraic conjugate of ![]() ![]() |
Step (6) | [SHOW MORE] | ||
8 | Every algebraic conjugate of ![]() ![]() |
Step (7) | [SHOW MORE] | ||
9 | The modulus of the algebraic norm of ![]() |
Steps (5), (8) | [SHOW MORE] | ||
10 | If the modulus of the algebraic norm of ![]() ![]() ![]() |
[SHOW MORE] | |||
11 | If the modulus of the algebraic norm of ![]() ![]() ![]() |
Steps (6), (8) | [SHOW MORE] | ||
12 | Either ![]() ![]() |
Steps (9), (10), (11) | Step-combination direct. |