By J. S. Lomont (Auth.)

ISBN-10: 1483231321

ISBN-13: 9781483231327

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**Extra info for Applications of Finite Groups**

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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.

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

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