Editing Perfect group (section)

== Examples ==
The smallest (non-trivial) perfect group is the [[alternating group]] ''A''<sub>5</sub>. More generally, any [[non-abelian group|non-abelian]] [[simple group]] is perfect since the commutator subgroup is a [[normal subgroup]] with abelian quotient. However, a perfect group need not be simple; for example, the [[special linear group]] over the [[field (mathematics)|field]] with 5 elements, SL(2,5) (or the [[binary icosahedral group]], which is [[group isomorphism|isomorphic]] to it) is perfect but not simple (it has a non-trivial [[center (group)|center]] containing <math>-\!\left(\begin{smallmatrix}1 & 0 \\ 0 & 1\end{smallmatrix}\right) = \left(\begin{smallmatrix}4 & 0 \\ 0 & 4\end{smallmatrix}\right)</math>).

The [[Direct product of groups|direct product]] of any two simple non-abelian groups is perfect but not simple; the commutator of two elements is [(''a'',''b''),(''c'',''d'')] = ([''a'',''c''],[''b'',''d'']). Since commutators in each simple group form a generating set, pairs of commutators form a generating set of the direct product.

The fundamental group of <math>SO(3)/I_{60}</math> is a perfect group of order 120.<ref>Milnor, John. "The Poincaré Conjecture." ''The millennium prize problems'' (2006): 70.</ref>

More generally, a [[quasisimple group]] (a perfect [[Central extension (mathematics)|central extension]] of a simple group) that is a non-trivial extension (and therefore not a simple group itself) is perfect but not simple; this includes all the [[soluble group|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''<sub>5</sub>). Similarly, the special linear group over the [[real number|real]] and [[complex number|complex]] numbers is perfect, but the general linear group GL is never perfect (except when trivial or over <math>\mathbb{F}_2</math>, 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 group|solvable]]; and 4 [[divisor|divides]] its [[order (group theory)|order]] (if finite), moreover, if 8 does not divide the order, then 3 does.<ref>Tobias Kildetoft (7 July 2015), [https://math.stackexchange.com/a/1357886/330413 answer] to [https://math.stackexchange.com/q/1357885/330413 "Is a non-trivial finite perfect group of order 4n?"]. ''Mathematics [[StackExchange]]''.  Accessed 7 July 2015.</ref>

Every [[acyclic group]] is perfect, but the converse is not true: ''A''<sub>5</sub> is perfect but not acyclic (in fact, not even [[Superperfect group|superperfect]]), see {{harv|Berrick|Hillman|2003}}. In fact, for <math>n\ge 5</math> the alternating group <math>A_n</math> is perfect but not superperfect, with <math>H_2(A_n,\Z) = \Z/2</math> for <math>n \ge 8</math>.

Any quotient of a perfect group is perfect. A non-trivial finite perfect group that 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 (algebra)|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 [[Issai Schur]] in 1904; it is isomorphic to the [[homology group]] <math>H_2(G)</math>.

In the '''plus construction''' of [[algebraic K-theory]], if we consider the group <math>\operatorname{GL}(A) = \text{colim} \operatorname{GL}_n(A)</math> for a [[commutative ring]] <math>A</math>, then the [[subgroup]] of elementary matrices <math>E(R)</math> forms a perfect subgroup.