By Peter Webb

This graduate-level textual content offers an intensive grounding within the illustration thought of finite teams over fields and jewelry. The booklet offers a balanced and complete account of the topic, detailing the tools had to learn representations that come up in lots of components of arithmetic. Key subject matters comprise the development and use of personality tables, the position of induction and limit, projective and easy modules for staff algebras, indecomposable representations, Brauer characters, and block conception. This classroom-tested textual content offers motivation via a number of labored examples, with workouts on the finish of every bankruptcy that try out the reader's wisdom, supply extra examples and perform, and contain effects now not confirmed within the textual content. must haves contain a graduate path in summary algebra, and familiarity with the houses of teams, earrings, box extensions, and linear algebra.

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**Example text**

We argue first that H is a normal subgroup. If the eigenvalues of ρ(h) are λ1 , . . , λn then, since these are roots of unity, |λ1 + · · · + λn | = n if and only if λ1 = · · · = λn . Thus |χ(h)| = χ(1) if and only if ρ(h) is multiplication by some scalar, and from this we see immediately that H is a normal subgroup. It also implies that H/ ker ρ is abelian. From Step 6 we see that H contains the non-identity element g, so simplicity of G forces H = G. Since ρ is not the trivial representation ker ρ = G, so simplicity of G again forces ker ρ = 1, so that G must be abelian.

A) If χ is a character of G, show that the function g → χ(xg) need not be a class function on G. 2 are orthogonal idempotents is equivalent to the validity of the following formulas, for all x ∈ G and for all of the simple characters χi of G: di |G| χi (g −1 )χj (xg) = g∈G 0 χi (x) if i = j, if i = j. 16. Show that the only group G which has character table a 5 b b2 4 4 b3 4 g |CG (g)| 1 20 χ0 χ1 χ2 χ3 χ4 1 1 1 1 1 1 1 i −1 −i 1 1 −1 1 −1 1 1 −i −1 i 4 −1 0 0 0 is G = C5 C4 = a, b a5 = b4 = 1, bab−1 = a2 .

Since all these matrices are square with independent rows or columns, they are invert−1 ible, and in fact (XC −1 )−1 = X T = CX . Therefore X T X = C. Another way to state the column orthogonality relations is r χi (g)χi (h) = i=1 |CG (g)| if g ∼ h, 0 if g ∼ h. This means that the column of the character table indexed by g ∈ G is orthogonal in the usual sense to all the other columns, except the column indexed by g −1 (because χ(g −1 ) = χ(g)), and the scalar product of those two columns is |CG (g)|.

### A Course in Finite Group Representation Theory by Peter Webb

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