isomorphism
An isomorphism says two structures are the same thing wearing different clothes. If you can match up their elements perfectly and the matching respects every operation, then anything true about one is true about the other — only the names of the elements differ. It is mathematics' formal way of saying “these are essentially identical.”
Formally, an isomorphism is a homomorphism that is also a bijection: it preserves the operations and pairs up the two structures' elements exactly one-to-one with nothing left over. Because it is invertible and its inverse is also a homomorphism, the relationship is perfectly symmetric — neither side is privileged. Two structures linked this way are called isomorphic.
Classifying objects “up to isomorphism” is a central goal of algebra: instead of an endless list of look-alike groups, you describe each genuinely different shape once. The hard part is often proving two structures are NOT isomorphic — for that you find an invariant property, like the number of elements, the number of elements of a given order, or whether the operation commutes, that one has and the other lacks.
The integers mod 4 under addition and the fourth roots of unity {1, i, -1, -i} under multiplication are isomorphic: match 1 with i, since adding repeatedly mirrors multiplying powers of i. Both are cyclic of order 4, the same structure in different clothing.
ℤ/4ℤ ≅ the 4th roots of unity under multiplication.
Equal size is necessary but not sufficient for isomorphism. There are two genuinely different groups of order 4 — the cyclic one and the Klein four-group — which have the same number of elements yet are not isomorphic, because their elements have different orders.