Talk:Maximal torus
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"Given a maximal torus T in G, every element g ∈ G is conjugate to an element in T." What if G is not compact? Does this hold iff g is semisimple? —The preceding unsigned comment was added by 84.165.180.8 (talk • contribs) 18:20, 23 December 2006 (UTC)
- No, the compact condition is necessary. Take SL(2,R) for instance. This group is semisimple, noncompact, with maximal torus equal to the rotation subgroup SO(2). Conjugacy must preserve the trace of a matrix and every matrix in SO(2) has a trace between −2 and 2. Therefore any matrix in SL(2,R) with |Tr| > 2 (the so-called hyperbolic elements) cannot be conjugate to a rotation. Likewise, the parabolic elements (the nonidentity elements with Tr^2 = 4) are not conjugate to rotations. -- Fropuff 01:55, 24 December 2006 (UTC)
