Perfect not implies semisimple

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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) need not satisfy the second group property (i.e., semisimple group)
View a complete list of group property non-implications | View a complete list of group property implications
Get more facts about perfect group|Get more facts about semisimple group


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).


The following is one recipe for constructing counterexamples. Let n \ge 5. Take the wreath product with base cyclic group:Z2 and acting group the alternating group A_n. This is a group of order 2^n(n!)/2. The center is cyclic of order two. Quotient out by it and get the inner automorphism group. This is a perfect group of order 2^{n-2}n! 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.