On the Holomorphisms of a Group (1918)(en)(2s) by Miller G. A.

By Miller G. A.

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By Miller G. A.

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Example text

C2. The one operation S' of C2 not contained in C1is the rotation by 180' about the vertical axis; ZS' is reflection in the horizontal plane through 0. Hence CzCl is the group consisting of the identity and of the reflection in a given plane; in other words, the group to which bilateral symmetry refers. The two ways described are the only ones by which improper rotations may be included in our groups. ) Hence this is the complete table of all finite groups of (proper and improper) rotations: I,, , DI,~~, D:,D:, (n = 2, 3, T, W , P; , , P; - a) WT.

1 goes into that with the coordinates xi, xi where + + + + + + + --t AB, and concepts logically defined in terms of them, we do affine geometry. I n affine geometry any vector basis el, e2 is as good as any other. The notion of the length Irl of a vector r transcends affine geometry and is basic for metric geometry. The square of the length of an arbitrary vector F is a quadratic form of its coordinates xl, x2 with constant coefficients gll, glz, g22. This is the essential content of Pythagoras' theorem.

I t is easy to see under what circumstances a linear transformation like (2) has an inverse, namely, if and only if its 2 ~ is different so-called modul ~ 1 1 -~ alza21 from 0. As long as we use no other concepts than those introduced so far, namely (1) addition b, (2) multiplication of a vecof vectors a tor a by a number A, (3) the operation by which two points A,B determine the vector + variables xl, x2 except forxl = x2 = 0 . There exist special coordinate systems, the Cartesian ones, in which this form assumes the simple x:; they consist of two muexpression x; tually perpendicular vectors el, es of equal length 1.

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