Iam not sure whether the following remarks answer the question.
Johann Heinrich Lambert claimed the existence of a primitive element
modulo the prime p, i. e., the cyclicity of the multiplicative group of
GF(p) in 1769. (Quoted after Dickson, History. Dover Publ. New York 1966)
Euler's attempt to prove it failed according to Gauss. (Euler, Novi
comm. acad. petr. 18 (1773) 85-135, 1774. Works II, 240-281. Gauss,
Disquisitiones, art. 56.) In the above paper, Euler coined the name
"primitive element."
Gauss gave a most beautiful proof for Lambert's claim. (Disquisitiones,
art. 52) What Gauss actually proved was a much more general theorem,
namely:
If G is a group such that the equation x^d = 1 has at most d solutions
for all d dividing |G|, then G is cyclic. Gauss not knowing the notion
of a group could not formulate the theorem.
Lambert's claim and Gauss' proof belong, of course, to the time of
proto-group-theory.
Cyclic symmetry and its use in algebra - and this does definitely not
answer the question - occurs in Fibonacci's liber abbaci. He
investigates certain systems of linear equation with a cyclic symmetry.
To distinguish the equations, he indexes them with certain of their
coefficients in the form
a, b, c, d
These letters stand for fractions. Manipulation of their denominators
and enumerators give the first solution. Then he replaces the above
string by
b, c, d, a
and says: Do the same thing to get the second solution, etc.!
Heinz Lueneburg