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4.24 (1973)

Partially Solved

Suppose $T$ is a non-abelian Sylow 2-subgroup of a finite simple group $G$.
$\qquad$ a) Suppose $T$ has nilpotency class $n$. The best possible bound for the exponent of the center of $T$ is $2^{n-1}$. This easily implies the bound $2^{n(n-1)}$ for the exponent of $T$, however, this is almost certainly too crude. What is the best possible bound?
$\qquad$ b) Is it possible that $T$ is the direct product of two proper subgroups?
$\qquad$ c) Is $T' = \Phi(T)$?
$\qquad$ d) Find a “small number” of subgroups $T_1, \dots, T_n$ of $T$ which depend only on the isomorphism class of $T$ such that $\{N_G(T_1), \dots, N_G(T_n)\}$ together control fusion in $T$ with respect to $G$ (in the sense of Alperin).

Contributor: D. M. Goldschmidt

Progress

b) Yes, it is. For example, the Sylow 2-subgroups of the alternating groups $A_{14}$ and $A_{15}$ are isomorphic to the Sylow 2-subgroup of $S_4 \times S_8$. The Sylow 2-subgroups of $D_4(q)$ for $q$ odd are also decomposable into direct products. (A. S. Kondratiev, Letter of October, 13, 1977.)
c) Not always; for example, for $G = \text{PSL}_3(q)$ with $q \equiv 1 \pmod 4$ (A. S. Kondratiev).

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