Past exam of the topology and groups course of the University of Oxford / 2023 / 2 / a / ii

Solution

F = G_{1} * G_{2}

, with its natural group homomorphisms

k_{j} : G_{j} \to F

, and form the normal closure

N = ⟨ ⟨ k_{1} (ϕ_{1} (h)) k_{2} (ϕ_{2} (h))^{- 1} : h \in G_{0} ⟩ ⟩ .

P = (G_{1} * G_{2}) / N,

j_{1}, j_{2}

obtained by composing

k_{1}, k_{2}

with the quotient map of groups. The defining relators ensure

j_{1} ϕ_{1} = j_{2} ϕ_{2} .

Thus the two prescribed images of every group element of

G_{0}

agree in

P

. This construction works for arbitrary group homomorphisms

ϕ_{1}, ϕ_{2}

; no injectivity is assumed, and

j_{1}, j_{2}

need not be injective. Its universal property is proved in the next part.

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