4. Let G be a finite group and 4 : G
element Z E FG to be
1 be a one dimensional representation. Define points
(a) Prove that Z is an idempotent.
G gEG [46-119. 2 141°3244 -
X= - -
(b) If R = FG, then show dimp Rx = 1.
5. Suppose the character table of a finite group G has the following two rows
K1 K2 Ki3 K4 K5 Kif K7
1 1 1 w² w w2 w
2 - 2 0 -1 - 1 1 1
where w = e2Tri/3 and the Ki denote conjugacy classes of G with K1 containing the ident
(a) Determine the remaining entries of the character table.
(b) Use part (a) to determine I G and the sizes of each conjugacy class.
This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.