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Unformatted text preview: 4.5. THE FULL SYMMETRY GROUP 235 group of a geometric ﬁgure is the group of g 2 SO.3; R/ that leave the
ﬁgure invariant. The full symmetry group is the group of g 2 O.3; R/ that
leave the ﬁgure invariant. The existence of reﬂection symmetries means
that the full symmetry group is strictly larger than the rotation group.
In the following discussion, I do not distinguish between linear isometries and their standard matrices.
Theorem 4.5.1. Let S be a geometric ﬁgure with full symmetry group G
and rotation group R D G \ SO.3; R/. Suppose S admits a reﬂection
symmetry J . Then R is an index 2 subgroup of G and G D R [ RJ . Proof. Suppose A is an element of G n R. Then det.A/ D
det.AJ / D 1, so AJ 2 R. Thus A D .AJ /J 2 RJ . 1 and
I For example, the full symmetry group of the cube has 48 elements.
What group is it? We could compute an injective homomorphism of the
full symmetry group into S6 using the action on the faces of the cube, or
into S8 using the action on the vertices. A more efﬁcient method is given
in Exercise 4.5.1; the result is that the full symmetry group is S4 Z2 .
Are there geometric ﬁgures with a nontrivial rotation group but with
no reﬂection symmetries? Such a ﬁgure must exhibit chirality or “handedness”; it must come in two versions that are mirror images of each other.
Consider, for example, a belt that is given n half twists (n 2) and then
fastened. There are two mirror image versions, with right–hand and left–
hand twists. Either version has rotation group Dn , the rotation group of the
n–gon, but no reﬂection symmetries. Reﬂections convert the right–handed
version into the left–handed version. See Figure 4.5.1. Figure 4.5.1. Twisted band with symmetry D3 . There exist chiral convex polyhedra with two types of regular polygonal faces, for example, the “snubcube,” shown in Figure 4.5.2 on the next
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 Fall '08
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 Algebra

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