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By J. S. Lomont (Auth.)

ISBN-10: 1483231321

ISBN-13: 9781483231327

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41 APPLICATIONS perfect square so that H could be written without the square root. , let us try putting pt + mlc* = (axpx + a2py + azpz + mQcß)\ (2:5a) where alt a2, a3, and ß are coefficients. Squaring the right side we get P* + m%* = ajpl + dip2, + atpl + τη\οψ + ( αι α 2 + α2αλ)ρχρν + ( α ^ + aza^)pxpt + (a 2 a 3 + (2:5b) a3a2)pypz + mQc{a^ + ßax)px + m0c(a2ß + ßa2)Py + m0c(azß + ßa3)pz, where we have assumed that the a's (and ß) commute with the p's, but not necessarily with each other. Clearly, our scheme will work if the a's and ß satisfy the following equations: a?

Matrix Groups Definition. A matrix group is a group in which the elements are square matrices (of the same dimension), the multiplication law is matrix multi­ plication, and the group inverse of an element is the matrix inverse. Example. The set of eight matrices ' - ( ί Ϊ) "·-(Ί -Ϊ) *-("i ? ) »-(-! "i)«■-(-? D M " Ó) used in previous examples is a matrix group. Thus, D2D5 = D7\ and, in fact, this group has the same multiplication table as the quaternion group Q (and is therefore isomorphic to Q).

If Γ is any irreducible representation of a group G, and χ is the corresponding primitive character, then χ must satisfy the following equation: r r r z d i jXiXj = ^Kj,krkXkk=l Proof. Substituting (2) of the preceding theorem in (1) we have 4. COMPUTATION OF CHARACTER TABLES 61 k=l Cancelling out / , and multiplying by d2 we get the described result. 4. Computation of Character Tables Two methods will be discussed here. (A) This method is based on the two equations k=l This method therefore requires (as does the second method also) the determina­ tion from the multiplication table of the quantities g, rit r, and h^ k.

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Applications of Finite Groups by J. S. Lomont (Auth.)

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