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Let $G$ be a finite group with a central subgroup $Z$ of order 2. Suppose that $Z$ has a direct factor in some (and hence any) 2-Sylow subgroup of $G$. Does this imply that $Z$ has a direct factor in $G$?

I have checked some families of examples I can discard. First, the result is true when $G$ is abelian, or more generally nilpotent, and even more generally when $G$ has a normal 2-Sylow. Second, having in mind that $G$ has even order, the result is very easy when $|G|=2$ (mod 4) and is also true when $|G|=4$ (mod 8). This reduces to the case when 8 divides $|G|$. Third, I also checked it when $|G|=2^np$ with $p$ prime. The first values for $|G|$ that are not covered are thus $72=8\times 9$, $120=8\times15$, $144=16\times9$... A simple argument also shows that a minimal counterexample has no nontrivial normal subgroup of odd order.

I also do not know in the case when $G/Z$ is simple. To discard the first few cases, for $G=\mathrm{SL}_2(\mathbf{F}_q)$, $q$ odd prime power, which has center of order 2, the center is not a direct factor in the 2-Sylow since $-I_2$ is square of an element of order 4.

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You meant to say "suppose that $Z$ has a complement in some/any 2-Sylow subgroup", right? Then you're looking for Gaschütz's theorem. It is usually discussed together with the Schur-Zassenhaus theorem and/or group cohomology in many introductory group theory books, for example Kurzweil-Stellmacher (theorem 3.3.2) or Aschbacher's book (theorem 10.4).

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  • $\begingroup$ Thanks! Let me state it fully (essentially copying as in math.ku.dk/~olsson/manus/Gruppe-2009/gruppe2007en_all.pdf, p16): let $M$ be a normal abelian $p$-subgroup of a finite group $G$. Then $M$ has a direct complement in $G$ if (and only if) it has a direct complement in some/any $p$-Sylow subgroup of $G$. $\endgroup$ – YCor Dec 23 '17 at 22:42

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