Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

~ 116]

ON ROOTS OF UNITY

239

permute the triangles in the following manner:
(t2t3t4), (tlt2) (t3t4), (tlt4) (t2t3).
Accordingly, since all the required groups contain a transformation corresponding to (tlt2)(t3t4), every such group must
contain a transformation XV, X belonging to (D). Hence,
if G contains (D) as a subgroup, it also contains V. If, however,
(C) were a subgroup of G, but not (D), then either V is contained in G, or else XV, where X belongs to (D) but not to
(C). In this event X may be written X1R, where Xi belongs
to (C). Hence finally, either V or RV belongs to G. However,
V2=(RV)2=R. Thus R, and therefore also V, are contained
in G in any case.
Again, if G contains a transformation corresponding to
(t2t3t4) or (tlt4(t2t3), such a transformation can be written XU
or XUVU-1, X belonging to (D). Hence, since G contains
(D) as we have just seen, it will contain either U or UVU-1
in the cases considered. We therefore have the following types:
(E) Group of order 36P generated by (C) as given in ~ 113:
Sl ==(1, C0, 2), T: xl=X'2, X2=X'3, X3=X'1,
and the transformation V of (3).
(F) Group of order 72f generated by Si, T, V and UVU-1.
(G) Group of order 216, generated by Si, T, V and U.
These groups are all primitive, and they all contain (D)
as an invariant subgroup. The group (G) is called the Hessian
group (cf. Jordan, Journal fur die reine und angewandte Mathematik, Bd. 84 (1878), p. 209).
116. On Roots of Unity. A solution of the equation
= 1,
n being a positive integer, is called a root of unity. A solution
a is in particular called a primitive nth root of unity, if n is the
least integer for which a = 1. In such a case n is called the
index of the root.
THEOREMS. 10. The product or ratio of two roots of unity
is again a root of unity.