- #1

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- Homework Statement:
- I'm confused about my notes about free groups, looking for help to understanding them.

- Relevant Equations:
- will put definitions below

Let ##S = \lbrace a, b \rbrace## and define ##F_S## to be the free group, i.e. the set of reduced words of ##\lbrace a, b \rbrace## with the operation concatenation. We then have the universal mapping property: Let ##\phi : S \rightarrow F_S## defined as ##s \mapsto s## and suppose ##\theta : S \rightarrow G## is any function where ##G## is a group. Then there exists unique homomorphism ##f : F_S \rightarrow G## such that ##\theta = f \circ \phi##.

For example, ##f(aba^{-1}) = f(a)(f(b)f(a)^{-1} = \theta(a)\theta(b)\theta(a)^{-1}##.

My question is, where does ##f## come from? It just seems like there is some step that I am missing to get ##f## in the first place?

For example, ##f(aba^{-1}) = f(a)(f(b)f(a)^{-1} = \theta(a)\theta(b)\theta(a)^{-1}##.

My question is, where does ##f## come from? It just seems like there is some step that I am missing to get ##f## in the first place?