# Perfect not implies semisimple

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., perfect group) neednotsatisfy the second group property (i.e., semisimple group)

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## Statement

It is possible for a group (in fact, a finite group) to be a perfect group (i.e., it equals its own derived subgroup) but not a semisimple group (i.e., it is not expressible as a central product of quasisimple groups, or equivalently, its layer is a proper subgroup of it).

## Proof

The following is one recipe for constructing counterexamples. Let . Take the wreath product with base cyclic group:Z2 and acting group the alternating group . This is a group of order . The center is cyclic of order two. Quotient out by it and get the inner automorphism group. This is a perfect group of order and is not semisimple.

The smallest example is the inner automorphism group of wreath product of Z2 and A5, which is a group of order 960. There is another similar example of order 960. 960 is the smallest order for any example.