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Contents 1 Statement 2 Particular cases 3 Related facts 4 Facts used 5 Proof 6 References 6.1 Textbook references

Statement

Let G be a finite group, and let d be the exponent of G: in other words, d is the least common multiple of the orders of all elements of G. Suppose k is a sufficiently large field for G: k is a field whose characteristic does not divide the order of G, and such that the polynomial x^d − 1 splits completely over k.

Then, k is a splitting field for G: Every linear representation of G that can be realized over an algebraic extension of k can in fact be realized over k.

Particular cases

Note that if the exponent is 2swhere s is odd, the existence of s^th roots guarantees the existence of (2s)^th roots. Hence, the smallest d to guarantee sufficiently large in such circumstances is taken as s.

The symmetric group of degree three is the first example where the d-values for sufficiently large and splitting diverge.

Group	Order	Smallest d such that existence of dth roots guarantees sufficiently large	Smallest d such that existence of dth roots guarantees splitting
trivial group	1	1	1
cyclic group:Z2	2	1	1
cyclic group:Z3	3	3	3
cyclic group:Z4	4	4	4
Klein four-group	4	1	1
cyclic group:Z5	5	5	5
symmetric group:S3	6	3	1
cyclic group:Z6	6	3	3
dihedral group:D8	8	4	1
quaternion group	8	4	4
symmetric group:S4	24	12	1

Related facts

• Sufficiently large implies splitting for every subquotient
• Splitting not implies sufficiently large
• Splitting field for a group implies splitting field for every quotient
• Splitting field for a group not implies splitting field for every subgroup

Facts used

1. Brauer's induction theorem (this is also called the characterization of linear characters lemma)

Proof

Given: A finite group G, a field k that is sufficiently large for G.

To prove: k is a splitting field for G.

Proof: By fact (1), every character of G over K is a -linear combination of characters induced from characters of elementary subgroups of G. Since elementary groups are supersolvable, every character of an elementary subgroup is induced from a linear character on some subgroup of it; hence, every character of G is a -linear combination of linear characters on subgroups.

Now, every linear character can be realized over k because k is sufficiently large, and the induced representation from a linear character can be realized over the same field, so there is a collection of representations realized over k whose characters have all the irreducible characters in their -span. This forces that all the irreducible representations over K can be realized over the field k.

References

Textbook references

• Linear representations of finite groups by Jean-Pierre Serre, 10-digit ISBN 0287901906 (English), ISBN 3540901906 (French), Page 94, Corollary to Theorem 24, Section 12.3, ^{More info}