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

214

THEORY OF EQUATIONS

Ex. 2. Show by ~~ 187, 188 that there can be no transitive Abelian
group of prime degree other than the cyclic group, and that there is no
irreducible Abelian equation of prime degree other than the cyclic
equation.
Ex. 3. Show that no transitive Abelian group of degree n can be of
lower order than n.
Ex. 4. Show that a transitive Abelian group of degree n is of the
order n. Weber, Vol. I, p. 578.
189. Solution of Abelian Equations. The solution of Abelian
equations may be reduced to the solution of cyclic equations.
In a transitive Abelian group every substitution, except the
identical one, involves all the elements and has the same number of elements in each cycle. Hence, if n is the total number
of elements and r is the number in one cycle, we must have
n = r ~ t, where t is the number of cycles in the substitution.
Let G be the group of an irreducible Abelian equation f(x)= 0,
and let s be any substitution except 1. If c, c1, *.., c,_ are the
cycles in s, we may write
S = CClC2.'. Ct_1.
Each of these cycles has for its elements r roots of the equation f(x)= 0. Hence we have
C (a '-* ",-, )
c,=((/,.../,_-),
C,_t1= (o-1... r,,-i),
where the a's, f's, *., o-'s are the roots of f(x) = 0.
Let s' be any substitution in the group G. We have, ~ 187,
S1-' S ~ Si = S.
The product si-lss, is obtained by performing upon each cycle
of s the substitution s,, ~ 88. As this operation leaves s as a