Perfect Group

A group that coincides with its commutator subgroup.

If G is a non-Abelian group, its commutator subgroup is a normal subgroup other than the trivial group. It follows that if G is simple, it must be perfect. The converse, however, is not necessarily true. For example, the special linear group SL_2(q) is always perfect if q>3 (Rose 1994, p. 61), but if q is not a power of 2 (i.e., the field characteristic of the finite field F_q is not 2), it is not simple, since its group center contains two elements: the 2×2 identity matrix I and its additive inverse -I, which are different because 1!=-1.

See also

Grün's Lemma

This entry contributed by Margherita Barile

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Holt, D. G. and Plesken, W. Perfect Groups. Oxford, England: Clarendon Press, 1989.Rose, J. S. A Course in Group Theory. New York: Dover, 1994.

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Perfect Group

Cite this as:

Barile, Margherita. "Perfect Group." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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