Which calculation finds the multiplicity of an irreducible character in a representation character?

• Conjugate the irreducible values, weight matched products by class size, sum and divide by |G|.
• Multiply matching values without conjugation or class weights, sum and divide by the number of classes.
• Conjugate the irreducible values, weight matched products by class size, sum and divide by the dimension.

S3 has three conjugacy classes. A proposed character table contains only the trivial and sign rows. A student says it is complete because both rows are irreducible. Is it complete?

Circle one:   True   False

The S3 sign character is (1,-1,1) on identity, transpositions, 3-cycles. Which set is its kernel?

• The identity and all three transpositions
• The identity and both three-letter cycles
• All transpositions and both three-letter cycles
• Only the two three-letter cycles, without identity

In a complex character table of a finite group, what does one row hold?

• One group element expressed in every possible basis
• One conjugacy class with every member written out
• One irreducible character across all the classes

The regular character of the cyclic group C3 is 3 at the identity and 0 elsewhere. What is the multiplicity of a nontrivial one-dimensional complex character?

Answer: ______________

An S3 representation has character (7,-1,1) on identity, transpositions, 3-cycles, of sizes 1,3,2. The sign row is (1,-1,1). What is its multiplicity?

• 2
• 1
• 0
• 3

An S3 representation contains two trivial copies, three sign copies and one standard copy. Their dimensions are 1,1,2. What is the representation's dimension?

Answer: ______________

For a complex representation of a finite group, chi(g)=chi(e) exactly on the identity and two other elements. What do those three elements form?

• A normal subgroup of the group
• A conjugacy class of three elements
• An irreducible character of degree three

Dan says S3's standard character (2,0,-1) contains two trivial copies because its identity entry is 2. Class sizes are 1,3,2. Which correction fits?

• The entry is the trivial multiplicity; no other class values are needed to justify the count.
• That entry is dimension; the weighted trivial multiplicity is 0.
• That entry is dimension; the weighted trivial multiplicity is 2.

On S3's classes of sizes 1,3,2, a class function f has values (1,1,-1). Its squared norm is (1+3+2)/6=1. A student concludes that f must be an irreducible character.

Circle one:   True   False