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## Homework Statement

Prove that if [tex]f: G \to H[/tex] and [tex]g: H \to K[/tex] are homomorphisms, then so is [tex]g \circ f: G \to K[/tex].

**2. The attempt at a solution**

Since [tex]f[/tex] is a homomorphism [tex](G, * )[/tex] and [tex](H, \circ)[/tex] are groups and [tex]f(a*b)= f(a) \circ f(b), \forall a,b \in G[/tex]. Likewise, [tex](K, +)[/tex] is a group and [tex]g(f(a) \circ f(b)) = g(f(a)) + g(f(b)), \forall f(a), f(b) \in H[/tex] with [tex]a, b \in G[/tex]. Hence, [tex]g \circ f = g(f(x))[/tex] [tex]\forall x\in G[/tex]. [tex]g \circ f[/tex] is a homomorphism.

Is this anything like correct?

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