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In mathematics, more specifically in the area of modern algebra known as group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian quotients (equivalently, its abelianization, which is the universal abelian quotient, is trivial). In symbols, a perfect group is one such that ''G''(1) = ''G'' (the commutator subgroup equals the group), or equivalently one such that ''G''ab = (its abelianization is trivial). == Examples == The smallest (non-trivial) perfect group is the alternating group ''A''5. More generally, any non-abelian simple group is perfect since the commutator subgroup is a normal subgroup with abelian quotient. Conversely, a perfect group need not be simple; for example, the special linear group SL(2,5) (or the binary icosahedral group which is isomorphic to it) is perfect but not simple (it has a non-trivial center containing ). More generally, a quasisimple group (a perfect central extension of a simple group) which is a non-trivial extension (i.e., not a simple group itself) is perfect but not simple; this includes all the insoluble non-simple finite special linear groups SL(''n'',''q'') as extensions of the projective special linear group PSL(''n'',''q'') (SL(2,5) is an extension of PSL(2,5), which is isomorphic to ''A''5). Similarly, the special linear group over the real and complex numbers is perfect, but the general linear group GL is never perfect (except when trivial or over F2, where it equals the special linear group), as the determinant gives a non-trivial abelianization and indeed the commutator subgroup is SL. A non-trivial perfect group, however, is necessarily not solvable; and 4 divides its order (if finite), moreover, if 8 does not divide the order, then 3 does. Every acyclic group is perfect, but the converse is not true: ''A''5 is perfect but not acyclic (in fact, not even superperfect), see . In fact, for ''n'' ≥ 5 the alternating group ''An'' is perfect but not superperfect, with ''H''2(''An'', Z) = Z/2 for ''n'' ≥ 8. Any quotient of a perfect group is perfect. A non-trivial finite perfect group which is not simple must then be an extension of at least one smaller simple non-abelian group. But it can be the extension of more than one simple group. In fact, the direct product of perfect groups is also perfect. Every perfect group ''G'' determines another perfect group ''E'' (its universal central extension) together with a surjection ''f:E'' → ''G'' whose kernel is in the center of ''E,'' such that ''f'' is universal with this property. The kernel of ''f'' is called the Schur multiplier of ''G'' because it was first studied by Schur in 1904; it is isomorphic to the homology group ''H2(G)''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「perfect group」の詳細全文を読む スポンサード リンク
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