Clifford's theorem
This fact is related to: linear representation theory
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Contents
Statement
Verbal statement
The restriction of any irreducible complex character of a group, to a normal subgroup, is a multiple of the sum of all conjugates in the whole group of some irreducible character of the normal subgroup.
Statement with symbols, using character-theoretic language
Let be a finite group and a normal subgroup of . Let be a complex irreducible character of and of such that:
Then:
where and denotes the character:
Further and are positive integers dividing the index . In fact, is the index of the subgroup , defined as:
is termed the inertial subgroup.
Further, divides the index .
Statement with symbols, using module-theoretic language
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]Particular cases
Conjugacy-closed subgroups
A conjugacy-closed subgroup is a subgroup such that any two elements of the subgroup conjugate in the whole group, are also conjugate in the subgroup. If is conjugacy-closed, then for any and thus, in this case, the restriction of the irreducible character from to is simply a multiple of .
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 70, Theorem 4.1 (formal statement, followed by proof).