Degree of irreducible representation divides index of abelian normal subgroup

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This article gives the statement, and possibly proof, of a constraint on numerical invariants that can be associated with a finite group

This article states a result of the form that one natural number divides another. Specifically, the (degree of a linear representation) of a/an/the (irreducible linear representation) divides the (index of a subgroup) of a/an/the (abelian normal subgroup).
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This fact is related to: linear representation theory
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Statement

For an algebraically closed field of characteristic zero

Let G be a finite group and \rho an irreducible representation of G over an algebraically closed field of characteristic zero. Let N be an Abelian normal subgroup (?) of G. Then the degree of \rho divides the index [G:N].

For a splitting field of characteristic zero

Let G be a finite group and \rho an irreducible representation of G over a splitting field for G of characteristic zero. Let N be an Abelian normal subgroup (?) of G. Then the degree of \rho divides the index [G:N].

Note that the two statements are equivalent because any representation of G that is irreducible over a splitting field is also irreducible over the algebraic closure.

For a splitting field, any characteristic not dividing the group order

Let G be a finite group and \rho an irreducible representation of G over a splitting field for G of any characteristic (which must not divide the order of G). Let N be an Abelian normal subgroup (?) of G. Then the degree of \rho divides the index [G:N].

This is equivalent to the other two formulations because degrees of irreducible representations are the same for all splitting fields.

Related facts

Similar divisibility facts

Further information: degrees of irreducible representations

Combinatorial/inequality facts

Converse

Opposite facts

Applications

  • Ito-Michler theorem: The easy direction of the theorem follows immediately from this statement. It basically states that if a finite group has an abelian normal p-Sylow subgroup, then p does not divide the degrees of any of its irreducible representations. The converse, which is the hard part of the Ito-Michler theorem, relies on much heavier machinery including the classification of finite simple groups.

Breakdown for a field that is not algebraically closed

Further information: cyclic group:Z3, linear representation theory of cyclic group:Z3

Let G be the cyclic group of order three and \R (real numbers) be the field. Then, there are two irreducible representations of G over \R: the trivial representation, and a two-dimensional representation given by the action by rotation by multiples of 2\pi/3. The two-dimensional representation has degree 2, and this does not divide the order of the group, which is 3.

This representation does not remain irreducible over \mathbb{C}: in fact, it decomposes as a direct sum of two irreducible representations, both of which have degree 1, which satisfies the condition of dividing the index of any abelian normal subgroup.

Examples

Note that the result is true for trivial reasons for a finite abelian group. We thus concentrate on non-abelian groups. In the table below, we list, for each group, the indexes of all subgroups that are maximal among abelian normal subgroups, the gcd of these indexes, the degrees of irreducible representations, and the lcm of these degrees. By the result stated, the lcm of degree of irreducible representations must divide the gcd of indices of all subgroups that are maximal among abelian normal subgroups. Note that in fact we do not need to take gcd -- we can take the minimum, because every finite group has an abelian normal subgroup whose order is divisible by the orders of all abelian normal subgroups.

Note that among the examples below, special linear group:SL(2,3) is the only case where the lcm of degrees of irreducible representations is strictly smaller than the gcd of index values for abelian normal subgroups.

Group Linear representation theory Indexes of maximal among abelian normal subgroups gcd of index values Degrees of irreducible representations lcm of degrees
symmetric group:S3 linear representation theory of symmetric group:S3 2 2 1,1,2 2
dihedral group:D8 linear representation theory of dihedral group:D8 2,2,2 2 1,1,1,1,2 2
quaternion group linear representation theory of quaternion group 2,2,2 2 1,1,1,1,2 2
dihedral group:D10 linear representation theory of dihedral group:D10 2 2 1,1,2,2 2
alternating group:A4 linear representation theory of alternating group:A4 3 3 1,1,1,3 3
symmetric group:S4 linear representation theory of symmetric group:S4 6 6 1,1,2,3,3 6
special linear group:SL(2,3) linear representation theory of special linear group:SL(2,3) 12 12 1,1,1,2,2,2,3 6
alternating group:A5 linear representation theory of alternating group:A5 60 60 1,3,3,4,5 60

Facts used

  1. Third isomorphism theorem
  2. Index is multiplicative
  3. Normality satisfies image condition
  4. Abelianness is quotient-closed
  5. Isotypical-or-induced lemma: Let G be a finite group, N a normal subgroup, and k an algebraically closed field of characteristic zero (remove those assumptions on k?). Then if \rho is an irreducible representation of G one of the following must hold:
    • \rho is induced from an irreducible representation of H, where H is a proper subgroup of G containing N
    • The restriction of \rho to N is an isotypical representation
  6. Normality satisfies intermediate subgroup condition
  7. degree of irreducible representation divides index of center: This result states that the degree of any irreducible representation must divide the index of the center, or equivalently, the order of the inner automorphism group.

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
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 prove the statement by a strong form of induction on the order.

Statement to be proved by induction on n: For any finite group of order n, and any abelian normal subgroup of the group, and any algebraically closed field k of characteristic zero, the degree of any irreducible linear representation of the finite group over k must divide the index of the abelian normal subgroup.

Our strong form uses that the statement is true for all smaller numbers.

Inductive hypothesis: The statement to be proved by induction is true for all m < n. In particular, it is true for all proper divisors of n.

Inductive goal:

Given: A group G of order n. An abelian normal subgroup N of G. An irreducible linear representation \rho of G over an algebraically closed field k of characteristic zero.

To prove: The degree of \rho divides the index [G:N].

Proof: Suppose \rho has kernel K. Let L = G/K.

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 L has an abelian normal subgroup NK/K whose index divides the index [G:N]. Facts (1), (2) N is abelian normal in G. [SHOW MORE]
2 \rho descends to an irreducible linear representation \rho' of L, having the same degree as \rho. \rho is irreducible, has kernel K [SHOW MORE]
3 If \rho is not faithful, i.e., K is nontrivial, then the degree of \rho divides [G:N]. Steps (1), (2) [SHOW MORE]
4 Either \rho is induced from an irreducible representation of H, where H is a proper subgroup of G containing N, or the restriction of \rho to N is isotypical. Fact (5) \rho is irreducible over an algebraically closed field (or splitting field) of characteristic zero, G is finite, and N is normal in G. Fact-direct
5 If \rho is induced from an irreducible representation \sigma of H, where H is a proper subgroup of G containing N, then \deg(\rho) = [G:H]\deg(\sigma) Definition of induced representation
6 If \rho is induced from an irreducible representation \sigma of H, we must have that \deg \sigma divides [H:N]. Fact (6) Step (5) (for notation) [SHOW MORE]
7 If \rho is induced from an irreducible representation \sigma of H, we obtain that the degree of \rho divides [G:N]. Fact (2) Steps (5), (6) [SHOW MORE]
8 If \rho is faithful and the restriction of \rho to N is isotypical, then N is contained in the center Z(G) of G. N is abelian. [SHOW MORE]
9 If \rho is faithful and the restriction of \rho to N is isotypical, then the degree of \rho divides [G:N]. Facts (2),(7) Step (8) [SHOW MORE]
10 In every case, the degree of \rho divides [G:N]. Steps (3), (4), (7), (9) [SHOW MORE]