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:No problem. As I mentioned in the edit summary, it appears to be a common confusion. Textbooks often explicitly prove A4 is not simple, and A5 is simple, and then An is simple for n≥5, but leave out A1, A2, A3 as being a little too small to say anything about. One might check out [[list of small groups]] to see how there are not very many ''small'' groups, so they can usually be checked by hand easily enough.
:The alternating group A<sub>3</sub> is a group of order 3. Every group of order 3 is a [[cyclic group]] of order 3. Every cyclic group of order 3 is isomorphic to A<sub>3</sub>. Sometimes one studies a particular cyclic group of order 3, the set {0,1,2} under addition modulo 3, also known as '''Z'''/3'''Z'''. This group is also isomorphic to A<sub>3</sub>. [[User:JackSchmidt|JackSchmidt]] ([[User talk:JackSchmidt#top|talk]]) 18:17, 31 December 2007 (UTC)
== [[Whitehead lemma]] steinberg etc. ==
Hi Jack,
I'm no expert on the group theory of the [[general linear group]]; I just stumbled into these from the [[unitary group]] → [[Steinberg group]]s, and suddenly found myself in (familiar!) [[algebraic K-theory]] land.
So I don't know what the derived subgroup of GL(''n'',''A'') looks like, for instance, or whether the group GL(''A'') determines the ring (all that comes to mind is that <math>Z(GL_n(A))=A^*</math>, which isn't enough but isn't bad). I'm not in a university any longer, so I'm not in touch with people who'd know (and I'm not really sure whom to ask even then).
As you've presumably noticed, I've added some discussion of the connection between the [[Steinberg group]] and the special linear group. Abstractly the answer is given by [[Steinberg_group_%28K-theory%29#K2|this short exact sequence]] relating St and GL via K<sub>1</sub> and K<sub>2</sub>; hopefully this answers your questions. (I don't know a presentation of SL in terms of transvections generally though.)
I do come from a [[geometric topology]] background, and before that [[algebraic geometry]], and have a long-standing affair with [[algebraic topology]], so I'm largely one of THEM, but I do like to understand what's going on (rather than quoting results), and I'm in no hurry, so I've the time to luxuriate (read: actually understand results).
I've added "stable" in [[Steinberg group]] where relevant; trust it looks saner now.
Isn't <math>\operatorname{E}_n(A)</math> always perfect (for <math>n\geq 3</math>), due to the 2nd Steinberg relation (this struck you as insane)?
(I have no intuition for why kernel St → GL should be the center, but then, I have no intuition for St period: it's purely formal to me.)
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