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text: 'A group isomorphism is a [-47t] which is [499 bijective].\nWe say that two groups are *isomorphic* if there is an isomorphism between them.\n\nIt turns out that isomorphism is a much more useful concept than true equality of [-3gd groups], and it captures the idea that "these two objects are the same group": the isomorphism shows us how to relabel the elements to see that they are indeed the same group.\n\nFor example, the trivial group is in some sense "the only group with one element", but it can be instantiated in many different ways: as $(\\{ a \\}, +_a)$, or $(\\{ b \\}, +_b)$, and so on (where $+_x$ is the [-3kb] taking $(x, x)$ to $x$).\nThey all behave in exactly the same ways for the purpose of group theory, but they are not literally identical.\nThey are all isomorphic, though: the map $\\{a \\} \\to \\{ b \\}$ given by $a \\mapsto b$ is an isomorphism of the respective groups.\n\nTwo groups are isomorphic if and only if they have the same [cayley_table Cayley table], possibly with rearrangement of rows/columns and with relabelling of elements.',
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