Quiz:Degrees of irreducible representations

Find the feasible or infeasible degrees of irreducible representations
{Which of the following is not a possibility for the multiset of the degrees of irreducible representations of a finite group over a splitting field of characteristic zero (such as the complex numbers)? - 1,1,1,1,2 + 1,1,1,1,3 - 1,1,1,1,4 - None of the above, i.e., they are all possibilities for the multiset of degrees of irreducible representations.
 * type=""}
 * Wrong. This occurs for both dihedral group:D8 (see linear representation theory of dihedral group:D8 and quaternion group (see linear representation theory of quaternion group). See also linear representation theory of groups of order 8.
 * Right. This is impossible because, by sum of squares of degrees of irreducible representations equals order of group, the order of the group is 13. But a group of order 13 must be cyclic of prime order, and hence must have all its degrees of irreducible representations equal to 1.
 * Wrong. This occurs for GA(1,5), the general affine group of degree one over field:F5.

{Which of the following is not a possibility for the multiset of the degrees of irreducible representations of a finite group over a splitting field of characteristic zero (such as the complex numbers)? - 1,1,2,3,3 - 1,1,1,2,2,2,3 - 1,1,1,1,2,2,2,2,2 + 1,1,1,1,1,1,1,2,2,3 - 1,1,1,1,1,1,1,1,2,2,2,2
 * type=""}
 * Wrong. This occurs as the degrees of irreducible representations for symmetric group:S4 (a group of order 24). See linear representation theory of symmetric group:S4. See also linear representation theory of groups of order 24.
 * Wrong. This occurs as the degrees of irreducible representations for special linear group:SL(2,3) (a group of order 24). See linear representation theory of special linear group:SL(2,3). See also linear representation theory of groups of order 24.
 * Wrong. This occurs as the degrees of irreducible representations for SmallGroup(24,8), a group of order 24. See also linear representation theory of groups of order 24.
 * Right. By sum of squares of degrees of irreducible representations equals order of group, the group has order 24. Further, we must have number of one-dimensional representations equals order of abelianization, which must divide 24. However, here, the number of 1s is 7, which does not divide 24. Thus, this does not occur as the degrees of irreducible representations of a finite group. See also linear representation theory of groups of order 24.
 * Wrong. This occurs as the degrees of irreducible representations of direct product of S3 and V4, a group of order 24. See also linear representation theory of groups of order 24.

{Which of the following is not a possibility for the multiset of the degrees of irreducible representations of a finite group over a splitting field of characteristic zero (such as the complex numbers)? - 1,1 - 1,1,2 - 1,1,1,3 - 1,1,1,1,4 + None of the above, i.e., they are all possibilities
 * type=""}
 * Wrong. This arises as the degrees of irreducible representations for cyclic group:Z2. See linear representation theory of cyclic group:Z2.
 * Wrong. This arises as the degrees of irreducible representations for symmetric group:S3, which can alternatively be viewed as the general affine group of degree one over field:F3. See linear representation theory of symmetric group:S3.
 * Wrong. This arises as the degrees of irreducible representations for alternating group:A4, which can alternatively be viewed as the general affine group of degree one over field:F4. See linear representation theory of alternating group:A4.
 * Wrong. This arises as the degrees of irreducible representations of GA(1,5), the general affine group of degree one over field:F5.
 * Right. (1,1) arises for cyclic group:Z2, the others all arise for the general affine group of degree one $$GA(1,q)$$ for $$q = 3,4,5$$ respectively. See linear representation theory of general affine group of degree one over a finite field.

Maximum and divisibility
For all the questions below, we consider irreducible representations over a splitting field of characteristic zero, such as the field of complex numbers.

{Which of the following statements is false in general about the degrees of irreducible representations of a finite group over a splitting field of characteristic zero? + The degree of any irreducible representation divides the index of any abelian subgroup in the group. - The degree of any irreducible representation is bounded by, but need not divide, the index of any abelian subgroup in the group. - The degree of any irreducible representation divides the index of any abelian normal subgroup in the group.
 * type=""}
 * Right. See degree of irreducible representation need not divide index of abelian subgroup
 * Wrong. See index of abelian subgroup bounds degree of irreducible representation.
 * Wrong. See degree of irreducible representation divides index of abelian normal subgroup.

{What is the largest possible value of the maximum degree of irreducible representation for a group of order 24 over a splitting field of characteristic zero (such as the complex numbers)? - 2 + 3 - 4 - 6 - 8
 * type=""}
 * See linear representation theory of groups of order 24.

{What is the largest possible value of the maximum degree of irreducible representation for a group of order $$2^{2n + 1}$$ over a splitting field of characteristic zero (such as the field of complex numbers) where $$n$$ is a positive integer? - 2 + $$2^n$$ - $$2^{n + 1}$$ - $$2^{2n - 1}$$ - $$2^{2n}$$ - $$2^{2n + 1}$$
 * type=""}
 * The maximum occurs for extraspecial groups, see linear representation theory of extraspecial groups. Obtaining this as an upper bound is easy: see order of inner automorphism group bounds square of degree of irreducible representation, and prime power order implies not centerless

{It is true in general that degree of irreducible representation divides index of abelian normal subgroup, when we are working with irreducible representations of finite groups over splitting fields of characteristic zero. Which of the following gives an example of a group where the least common multiple of the degrees of irreducible representations is strictly smaller than the greatest common divisor of the index values of all abelian normal subgroups? - symmetric group:S3 - symmetric group:S4 - alternating group:A4 + special linear group:SL(2,3) - alternating group:A5 - special linear group:SL(2,5)
 * type=""}

Groups of prime power order: smallest counterexamples
{What is the smallest power $$2^n$$ such that there exist two groups of order $$2^n$$ of the same nilpotency class but with different multisets of degrees of irreducible representations over the field of complex numbers? - 8 - 16 + 32 - 64 - 128
 * type=""}
 * See degrees of irreducible representations need not determine nilpotency class

{What is the smallest power $$2^n$$ such that there exist two groups of order $$2^n$$ with the same multiset of degrees of irreducible representations over the field of complex numbers but such that the conjugacy class size statistics are different for the groups? - 8 - 16 - 32 + 64 - 128
 * type=""}
 * See [[linear representation theory of groups of order 8
 * See linear representation theory of groups of order 16
 * See linear representation theory of groups of order 32
 * See degrees of irreducible representations need not determine conjugacy class size statistics. See also linear representation theory of groups of order 64
 * See [[linear representation theory of groups of order 128

{What is the smallest power $$2^n$$ such that there exist two groups of order $$2^n$$ with the conjugacy class size statistics but such that the degrees of irreducible representations are different for the groups? - 8 - 16 - 32 - 64 + 128
 * type=""}
 * See linear representation theory of groups of order 8
 * See linear representation theory of groups of order 16
 * See linear representation theory of groups of order 32
 * See linear representation theory of groups of order 64
 * See conjugacy class size statistics need not determine degrees of irreducible representations, see also linear representation theory of groups of order 128