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Let $G$ be a finite group, a subgroup $K$ is called quasisimple if $K$ is perfect and $K / Z(G)$ is simple, and it is called a component if it is quasisimple and subnormal. Denote by $E(G)$ the group generated by all components of $G$ (sometimes called the layer of $G$).

Is it possible for $E(G)$ to be a $2$-group, or is this not possible?

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It can be shown that the components of a finite group commute with each other, and so $E(G)$ is a central product of quasisimple groups, and is perfect. So $E(G)=1$ is possible, but it cannot be a nontrivial $2$-group.

More generally, it is not hard to prove that any group that is generated by perfect subgroups is perfect, and hence $E(G)$ is perfect.

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