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A group isomorphism is a bijective homomorphism. Two groups are isomorphic if there exists an isomorphism between them, meaning they have identical algebraic structure. Isomorphic groups differ only in notation.
You've already studied homomorphisms — maps φ: G → H that preserve the group operation, meaning φ(ab) = φ(a)φ(b). A homomorphism is the group-theoretic notion of "structure-preserving map." An isomorphism is a homomorphism with one additional requirement: bijectivity. The map must be one-to-one (injective) and onto (surjective). When such a map exists, we write G ≅ H and say the groups are *isomorphic*.
The intuition is that isomorphic groups are the same group wearing different clothes. Consider the group of integers under addition modulo 4 ({0,1,2,3} with operation +₄) and the group of rotations of a square ({0°, 90°, 180°, 270°} with operation "followed by"). Both have order 4, both have a generator that cycles through all elements, and the structure of their Cayley tables is identical. The map φ(k) = rotation by 90k degrees is a bijective homomorphism — an isomorphism. In abstract algebra, these two groups are indistinguishable; any theorem proved for one holds for the other.
Not all groups of the same order are isomorphic. Consider two groups of order 4: ℤ₄ (cyclic, with a single generator of order 4) and the Klein four-group V₄ (where every non-identity element has order 2, i.e., a² = e for all a). These have the same size but different algebraic structure: ℤ₄ has an element of order 4, V₄ does not. No bijection between them can preserve the group operation, so V₄ ≇ ℤ₄. To prove two groups are *not* isomorphic, you find a structural invariant — a property preserved by any isomorphism, like the order of elements, the number of elements of each order, or commutativity — that differs between the two groups.
Isomorphisms are the equivalence relation of group theory: they tell you when two groups are "the same" at the level of abstract structure. The First Isomorphism Theorem, which you'll study next, makes this even more powerful by connecting isomorphisms to quotient groups — showing that every homomorphism "factors through" an isomorphism in a canonical way.