It is an exercise in Kargapolov to prove that a set G with a binary operation · is a group, if

  1. the operation is associative;
  2. the operation guarantees left and right quotients; i.e. for each pair of elements a, b in G there are G-elements x, y—called respectively left and right quotients of b by a—such that ax = b, ya = b.

(R+, · ) ≅ (R, +) via x ↦ ln x, but (by M. A. Armstrong's Groups and Symmetry (1988), pp.34–5) (Q, +) ≇ (Q+, ×), for were φ : Q → Q+ an isomorphism, then for some x in Q we'd have 2 = φ(x) = φ(x/2 + x/2) = φ(x/2)2, which is impossible because φ(x/2) is rational and √2 is not.

Three exercises (1.2.2–1.2.4):

  1. If an = e then |a| divides n.
  2. If ab = ba then |ab| = |a|·|b|.
  3. If |a| = m and |b| = n then G contains an element—not always ab—whose order is lcm(m, n).

Proposition. If a, bG then |ab| = |ba|.

Proof. If |ba| < |ab| then 1 = (ab)⋯(ab)
(|ab| times)
= a(ba)⋯(ba)b
(⩾ |ba| times)
= (ab)⋯, contradicting the minimality of the order of |ab|. ▮

Alternate proof. One may note that 1 = (ab)|ab|b−1a−1 = (ab)|ab|−1, so that (ba)|ab| = b(ab)|ab|−1a = bb−1a−1a = 1, whence |ba| ⩽ |ab|. Reversing the roles of a and b establishes the result. ▮