Suppose that φ is an isomorphism from a group G onto another group . Then

(1) φ carries the identity of G to the identity of .

(2) For every integer n and for every element a of G, .

(3) For elements a and b in G, a and b commute if and only if φ(a) and φ(b) commute.

(4) |a| = |φ(a)| for all a in G (isomorphisms preserve order).

For a fixed integer k and a fixed group element b in G, x^k = b has the same number of solutions in G as does the equation x^k = φ(b) in .