By J. S. Lomont (Auth.)
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For the quaternion matrix group, the sum on the left becomes |(2x2-6x2) = -1, so the group is of the second kind. Since matrix groups of the first or second kind are equivalent to their complex conjugates, it follows that their characters must be real functions. Further, it follows from our theorems on the equivalence of matrices that every matrix of a group of second kind is equivalent to a real matrix (but the same transformation matrix need not transform every matrix of the group to real form).
We come now to the study of a subgroup which will be of considerable use in representation theory. By a commutator (in a group) is meant a group element which can be expressed as a product of group elements in the form Α~λΒΛΑΒ. The commutator which was used for matrices is not used here because groups have no operation of addition. However, what we were usually interested in was whether or not the commutator of two matrices was zero. But, if Dx and D2 are nonsingular, then [Dlt D2] = 0 if and only if Dx D2 D^D^1 = I.
30 II. GROUPS I D, 1 2 3 4 5 2 3 / 1 5 4 1 5 4 2 3 / 3 4 5 / 1 2 4 2 1 5 / 3 5 / 3 4 2 1 Clearly, the elements / , alt and
Applications of Finite Groups by J. S. Lomont (Auth.)