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I recently realized for the first time (while teaching my undergraduate abstract algebra course) that simple groups and fields share the analogous property that neither has (nontrivial) factor groups/rings. Likewise, any homomorphism from a simple group or a field is either the 0 map or an embedding (which of course is just a corollary to my previous statement).

1) Has this relationship ever served as motivation for some line of research?

It makes sense that $\mathbb{Z}_p$ is both a simple group and a field.

2) Are there groups analogous to larger fields, say, $\mathbb{Q}$ or $\mathbb{Z}_p[\sqrt{q}]$?

3) On the other hand, there are nonabelian simple groups and, of course, the sporadic groups. Are there fields (or types of fields) then corresponding to the various types of simple groups?

Or maybe this is all just a coincidence and/or dead-end?

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What do you mean by the second example in 2. A (or rather the) quadratic extension of field with p elements or something else? –  quid Mar 13 '13 at 14:13
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There are also simple rings, simple modules, simple Lie algebras, and general simple objects in categories (especially in varieties of algebras) which are objects with no nontrivial quotients. A priori, you might as well open the field (no pun intended) to all such things. –  Todd Trimble Mar 13 '13 at 14:40
    
I agree with Todd. The main observation here is that simple commutative groups and simple commutative rings become rather boring objects (from the point of view of group theory and ring theory, respectively), but simple groups and simple rings (non-commutative ones) are very interesting. –  Tom De Medts Mar 13 '13 at 15:06
    
@quid: Yes, the quadratic extension. I was thinking (in my two examples) about non-finite fields (hence the rationals) and extension fields (hence the quadratic extension). –  Aeryk Mar 13 '13 at 16:02
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@Aeryk: (1) the paradigmatic such case would be the classification of simple complex (ha ha) Lie groups as a motivation to study the classification of simple complex Lie algebras. (2) There are clear cases (as in Morita equivalence) that set up correspondences between simple objects. But insofar as quotient objects (for varieties of algebras) correspond to internal congruence relations, it may be more or less hard to understand simple objects obviously depends on how hard or easy it is to detect congruence relations. At this level of generality, I don't think I can say much more. –  Todd Trimble Mar 13 '13 at 17:56

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