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Lemma·L37

Homomorphisms preserve identity, inverses, and subgroups

A homomorphism sends the identity to the identity, inverses to inverses, and its image is a subgroup.

Let be a homomorphism of groups. Then (i) ; (ii) for every ; (iii) the image is a subgroup of .
In words
A homomorphism carries the identity of the first group to the identity of the second, carries each inverse to the inverse of the image, and its set of output values is a subgroup of the target.
Rests onF02F05F09F13F14A02
Never needed: F03 · F04 · F06 · F08 · F10 · F11 · F12 · A01 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).

Proof

  1. 1
    (i) From (D038 (G2)) and the homomorphism property, . Also ((G2) in ). Comparing, , and left cancellation L28 (iii) gives .
  2. 2
    (ii) Fix . Then by the homomorphism property, D038 (G3), and (i); symmetrically . So is an inverse of , and by uniqueness of inverses L28 (ii), .
  3. 3
    (iii) Verify the subgroup clauses for (D007). Identity: by (i). Closure: if and then (homomorphism property). Inverses: by (ii). Hence .
∎

Remarks

So a homomorphism preserves the entire group skeleton, not just products. Part (iii) says the image is a group in its own right; combined with the kernel, it sets up the first isomorphism theorem , provable once quotient groups are built.
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L38L39
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