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

CHAPTER VI

GROUPS HAVING SIMPLE ABSTRACT DEFINITIONS
56. Groups Generated by Two Operators Having a Common
Square. If Si, S2 represent two operators of order 2 they evidently satisfy the equation S12 =S22, and the cyclic group (51S2)
generated by SlS2 is invariant under Si and s2. In fact, sis,
is transformed into its inverse by each of the two operators
sli S2, and hence (sl, s2) is the dihedral group whose order is
the double of the order of sIS2, as has been observed in ~ 26.
Hence the equations
S12= S22 =, (SS2)= 1
serve as a complete definition of the dihedral group of order
2n, if we assume that the order of SlS2 is exactly n. Throughout the present chapter it will be assumed, unless the contrary
is stated, that the condition s1"= implies that the order of
Si is exactly n. This fact is sometimes expressed by saying
that Si fulfils the condition sl =l, while the statement sl
satisfies the condition s1" =1 may imply merely that the order
of Si is a divisor of n.* We shall employ the terms fulfil and
satisfy with these meanings throughout the present chapter,
so that the equation sl = 1 implies that Si fulfils this condition
unless the contrary is stated.
If si, S2 are any two operators which satisfy the equation
S12 =S22,
they generate a group G under which si2 is invariant. That
is, the cyclic group generated by s12 is composed of operators
which are invariant under G, since s12 is commutative with each
* Quarterly Journal of Mathematics, vol. 41 (1909-10), p. 169.
143