= Group isomorphism
{title2=$\varphi:G\xrightarrow{\sim}H$}
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A <group isomorphism> is a <bijective> <group homomorphism> $\varphi:G\to H$. It matches the entire <group operation>: $\varphi(ab)=\varphi(a)\varphi(b)$, with each product taken in its own <group>. Its inverse is automatically a <group homomorphism>. A <group automorphism> is the special case $H=G$.<Group Theory (v4.01) by James S. Milne>[Definition 1.20]{ref}
The target is part of the claim. The doubling example under <isomorphism> is a <group isomorphism> $d:\mathbb Z\to2\mathbb Z$, between the additive <integers> and the additive <even integers>. Exactly the same rule $d(k)=2k$, declared as $d:\mathbb Z\to\mathbb Z$, is an <injective> <endomorphism> that misses $1$ and every other <odd integer>, so is not a <group automorphism>. Changing the declared target changes <surjectivity>, even though it does not change the formula or its compatibility with <addition>.
An <isomorphism> lets a calculation in one model solve the same problem in another. The example under <group> identifies <addition> modulo four with <function composition> of <rotations> of a <square> via $k\mapsto r^k$. Through this <group isomorphism>, adding exponents corresponds exactly to composing motions. More generally, a <group isomorphism> preserves <order of a group element>, carries <subgroups> to <subgroups>, and carries a <generating set of a group> to a <generating set of a group>. Having equally many elements is insufficient: the <permutation> in <group automorphism> already fails compatibility on two inputs.
Why does the inverse preserve the <group operation>? Write $u=\varphi(a)$ and $v=\varphi(b)$, possible because of <surjectivity>. Then
$$
\varphi^{-1}(uv)=\varphi^{-1}(\varphi(a)\varphi(b))=\varphi^{-1}(\varphi(ab))=ab=\varphi^{-1}(u)\varphi^{-1}(v).
$$
This proves the inverse condition directly. Analogous calculations work for each operation of a <ring> and for <addition> and <scalar multiplication> in a <vector space>. The argument relies on preserving those operations; it supplies no general continuity claim for the inverse of a <continuous> <bijection> between <topological spaces>.
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