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Here we study the group-theoretic concept of normality by characterizing normal subgroups as they appear within a parent group's Cayley graph. Vertex colorings induce certain combinatorially-defined graph quotients which elucidate the existence or non-existence of normal subgroups. We use this to prove the simplicity of the alternating groups on 5 or more letters. Through vertex-colorings we obtain ... read moreembeddings of normal subgroups within their parent group; studying the deck transformation group of certain covers by Cayley graphs provides a presentation for these normal subgroups. Combining the combinatorial and topological idea, we finally show that non-trivial finitely generated normal subgroups of finite rank free groups are finite index.read less
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