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If F and H are finite such that ker(G →* H) ≤ im(F →* G), then G is finite.
F
H
ker(G →* H) ≤ im(F →* G)
G
If F and H are finite such that ker(G →+ H) ≤ im(F →+ G), then G is finite.
ker(G →+ H) ≤ im(F →+ G)
If F and H are finite such that ker(G →* H) = im(F →* G), then G is finite.
ker(G →* H) = im(F →* G)
If F and H are finite such that ker(G →+ H) = im(F →+ G), then G is finite.
ker(G →+ H) = im(F →+ G)
If ker(G →* H) and H are finite, then G is finite.
ker(G →* H)
If ker(G →+ H) and H are finite, then G is finite.
ker(G →+ H)
If F and coker(F →* G) are finite, then G is finite.
coker(F →* G)
If F and coker(F →+ G) are finite, then G is finite.
coker(F →+ G)