# Artin's induction theorem

This article states an induction theorem: a result relating the linear characters and linear representations of a group with the characters/representations induced from the linear characters/representations of subgroups

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This fact is related to: linear representation theory

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## Contents

## Statement

Let be a finite group and a family of subgroups of . Then the following are equivalent:

- The union of conjugates of elements of cover the whole of
- Every character of over is a rational linear combination of characters induced from characters of members of

Further, these equivalent conditions hold if is the collection of all cyclic subgroups of .

## Related facts

## Facts used

## Proof

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### Proof idea

With a little linear algebra, we can show that if a character of is a complex linear combination of characters induced from members of , all the coefficients are in fact rational. Thus, the problem reduces to showing that the class functions induced from members of span the space of all class functions on .

This proof follows by using Frobenius reciprocity, and the fact that the only class function on which restricts to the zero function on every member of , is the zero function on the whole of .

### Proof details: (1) implies (2)

**Given**: A finite group with a family of subgroups such that the union of conjugates of members of is .

**To prove**: Every character of is a rational linear combination of characters induced from characters of members of .

**Proof**:

Step no. | Assertion/construction | Given data used | Facts used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | The (rational) vector space span of the irreducible characters contain all the characters. | [SHOW MORE] | |||

2 | A subset of the -span of characters with the property that its -span contains all class functions also has the property that its -span contains all characters. | Something about linear algebra; fields are linearly closed |
(1) | ||

3 | To show what we need to prove, it suffices to show that every class function is a -linear combination of characters induced from members of . | (1), (2) | [SHOW MORE] | ||

4 | To show what we need to prove, it suffices to show that every class function is a -linear combination of class functions induced from members of . | (3) | [SHOW MORE] | ||

5 | Let be the -span of all the class functions of . is also the space of class functions of . Let be the span of all class functions induced from characters of members of . In other words: . | ||||

6 | Let be the orthogonal complement to in with respect to the inner product of class functions: | (5) | |||

7 | Suppose . Then, for any and any class function of , we have: . | Fact (1) (Frobenius reciprocity) | |||

8 | (5), (6) | [SHOW MORE] | |||

9 | (7), (8) | [SHOW MORE] | |||

10 | is orthogonal to every class function of . | (9) | [SHOW MORE] | ||

11 | for every | The inner product is an inner product. |
(10) | [SHOW MORE] | |

12 | for every element of in the union of conjugates of members of . | (6), (7): is a class function; (11) |
[SHOW MORE] | ||

13 | is identically the zero function | is the union of all conjugates of members of | (12) | [SHOW MORE] | |

14 | (13) and (7) | [SHOW MORE] | |||

15 | , i.e., every class function on is in the -span of those induced from class functions on members of . | (14), (5), and (6) | [SHOW MORE] | ||

16 | We are done. | (4) and (15) | Follows directly. |

### Proof details: (2) implies (1)

The proof here is essentially the same; it uses Frobenius reciprocity to reason in the opposite direction.

**Given**: A finite group with a family of subgroups such that every character of is a rational linear combination of characters induced from .

**To prove**: is the union of conjugates of members of .

**Proof**: Since every character of is a rational linear combination of the characters induced from , it is in particular true that the -span of class functions induced from class functions of , is the whole space of class functions on .

Taking the usual inner product of class functions:

.

Now, suppose is a class function of that takes the value on the union of conjugates of and is outside. Then we have that for every and every class function of :

.

By Frobenius reciprocity, we get:

.

In other words, is orthogonal to all the class functions induced from members of . By assumption, is thus orthogonal to every class functino of , forcing . By te way we defined , we obtain that the union of conjugates of must be the whole group .

### Proof details for the additional observation

The additional observation follows from fact (2): every group is a union of cyclic subgroups.