As in corollary 9, we see that a subpregroup (Q,E) of (P,D) is a subgroup of (P,D)
= E=QxQcD. A slightly less obvious fact is
10. Corollary: Let xeP. The following are equivalent: (i) x is in a subgroup of P,
(iii) (x,x)eD, (iv) Let x be the cyclic subgroup generated by x in
U(P). Then x is a subgroup of P.
Proof: Use corollary 8 to show (i) = (ii) = (iii). Since (iv) implies (i), it suffices
to prove (iii) = (iv). Suppose (x,x)D. Then
for all /, 1 / n. Then (xn~\x)D and (x,xn~l) by corollary 8, so (x,xn~\x,x)D,
whence axiom (P4) implies
xn + leP.
From (P2) we see that xCP. Thus x is
a subgroup of P by corollaries 8 and 9. •
The elements of a pregroup fall naturally into two sets: those contained in sub-
groups of P and those not contained in subgroups of P.
11. Definition: Let P be a pregroup. An element xeP contained in a subgroup of P
is called a cycfe element of P. If x e f is not cyclic, then it is simple. A pregroup
with no non-trivial cyclic element is called a simple pregroup.