An introduction to the modern theory of equations, by Florian Cajori.

S.UBSTITUTION-GROUPS

127

V is a required function. By inspection we see that ~ is composed of
parts which are themselves functions of the kind sought for. These parts are
- (ea + a3)(tcl + tc3) - (a2 + (4) (t2 4- tC4),
((t,2 + C(32)C1c3 - (C22 + a42)C2C4,
aCla3(C12 + C32) + t2at4(C22 + C42).
'For t = 1, cl = c = - 1 and C2 = C4 = 1 we obtain the simpler form
(Cl + (4X- t -2 - (4.
For t = 0, ci = C3 = 1, C2 = C4 - i, we obtain the simpler forms
Cta32+ (t32 - t22 - a42,
el(t3 - a2a4.
Ex. 2. Assuming p = cc - (a2 + i(tx, derive functions which belong to
G3(3) as a sub-group of G6(3).
Taking t = 0, we get (i - 2)(at(X32 + (at22 +- a2ea12) + (i + 2)(a2(a32
+ aast12 + (X1(2 2). Then show that at1a32 + (x- 322 + a2ai2 and a2a32
+ (X3(2 + Cia(22 each belong to G3(3).
Ex. 3. Find the group to which ((al +- c3)(a(2 + a4) belongs.
We find, by trial, which of the substitutions of the symmetric group of
the fourth degree leave the function unaltered. These substitutions are
1, ((t12)(a3a4), (tlea3)(Ct2a4), (a(t4)(a2ea3), ((1g3), (at2at4), (a12a3at4),
((tx1(at4g2). These substitutions constitute the required group. From
~ 10i it is seen to be G8(4). From the behavior of this group toward the
given function, show that the group is imprimitive.
Ex. 4. Find the group to which Cr1ia2 + a3a4 - (c1a3 + a2a4) belongs.
Ex. 5. Find the group to which (a e- tl)((t2 - (3) belongs.
Ex. 6. Find the group to which (ala2 - a(3(4)2(al (t3 +- C2a4)2 belongs.
Ex. 7. Prove that the substitutions which leave unaltered a function
of n distinct letters, form together a group of the nth degree.
* Ex. 8. Show that (aclt2q 4- 2pa3q - + +.. a,-Pa q + c,,Pai where p
and q are distinct positive integers, is a cyclic function.
Ex. 9. By inspection show that {(a - (t2) + i (ai - t3)}2 belongs to
G2(4) as.a sub-group of G24(4). Compare with Ex. 1.
Ex. 10. Show that the cross-ratio of four points (~ 78) k = AC AD
BC BD'
when k is not equal to - 1 or to w. is a function which.belongs to G4(4)II;