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Let G be a group. For $g \in G$ write $l_g : G \rightarrow G$ for the function $l_g(g') = gg'$. Then $l_g$ is a permutation of G. Moveover, if we define $\Lambda: G \rightarrow S(G)$ by $\Lambda(g) = l_g$, then $\Lambda$ is an embedding. In consequence, if G is finite, then it is isomorphic to a subgroup of $S_{|G|}$.

My question:

What does "embedding" mean? Does it have a formal definition, or does it just mean a function that is not surjective? I tried to search the definition online but couldn't find anything useful.

Thanks in advance

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2 Answers 2

up vote 3 down vote accepted

Here it means an injective homomorphism.

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I was just wondering...since you said "Here it means...", do you mean that there is no formal definition of embedding? So can it mean something else in another context? –  user58289 Feb 13 '13 at 23:18
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@Artus: What I meant is that embeddings exist in other fields of mathematics. So, here the homomorphism is a group homomorphism, but in other contexts it may be a homomorphism of a different kind of objects. –  user3533 Feb 15 '13 at 20:01

In this case, embedding is being used to say that $G$ has a copy of itself into $S(G)$, that is, there exists an injective homomorphism onto its image.

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