A group that coincides with its commutator subgroup.

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

## See also

Grün's Lemma
*This entry contributed by Margherita
Barile*

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## References

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.## Referenced on Wolfram|Alpha

Perfect Group
## Cite this as:

Barile, Margherita. "Perfect Group." From *MathWorld*--A Wolfram Web Resource, created by Eric
W. Weisstein. https://mathworld.wolfram.com/PerfectGroup.html

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