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Every way to tell if two groups are isomorphic

Build an explicit bijection to prove it -- find one mismatched invariant to disprove it instantly.

Check the cheap invariants first, in order: order, then abelian-ness, then cyclic-ness, then the full element-order multiset. A single mismatch rules out isomorphism completely, with no bijection ever constructed. If every invariant matches, that's still not a proof -- matching invariants only means a bijection might exist, so the next step is to actually build one (or run a real search for one).

Proving two groups ARE isomorphic: build the explicit map

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A bijective homomorphism f: G \to H; when one exists, G and H are called isomorphic, written G \cong H, meaning they are structurally identical.

Example

f:\mathbb{Z}_4\to U(5), f(k)=2^k\bmod 5, is a bijective homomorphism, so \mathbb{Z}_4\cong U(5).

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Define f:G\to H, confirm f(ab)=f(a)f(b) for every pair, and confirm f is both injective and surjective -- or, when G and H are finite of equal size, prove injective alone and get surjective for free.

Disproving isomorphism: check invariants, cheapest first

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The number of elements in a group, written |G| -- finite if that count is finite, infinite otherwise.

Matching order is necessary for an isomorphism, but never sufficient by itself -- |G|=|H| still leaves every other invariant to check.

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A group in which the operation commutes: a*b = b*a for every pair of elements.

An isomorphism preserves commuting, so G abelian if and only if H abelian -- one non-abelian partner instantly rules it out.

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A group in which every element is a power of a single element -- that one element generates the whole group.

An isomorphism preserves single-generator structure, so G cyclic if and only if H cyclic.

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The smallest positive integer n with a^n = e, if one exists; the element has infinite order otherwise.

Example

\mathbb{Z}_4's element orders are 1,4,2,4; U(8)'s are 1,2,2,2. \mathbb{Z}_4 has an element of order 4 and U(8) has none, so \mathbb{Z}_4\not\cong U(8).

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The full multiset of element orders must match exactly between G and H -- not just the largest or smallest one.

When every invariant matches, search for a bijection directly

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