algebra / Group theory
For an extension $G = A \rtimes B$ of elementary abelian $p$-groups with $a \in A$ satisfying $C_B(a) = 1$, must $H = \langle a, B\rangle$ satisfy $\operatorname{rank}(Z(H) \cap H') \le \operatorname{rank}(B)$? An explicit extension violates the bound.
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For an extension $G = A \rtimes B$ of elementary abelian $p$-groups with $a \in A$ satisfying $C_B(a) = 1$, must $H = \langle a, B\rangle$ satisfy $\operatorname{rank}(Z(H) \cap H') \le \operatorname{rank}(B)$? An explicit extension violates the bound.
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For an extension $G = A \rtimes B$ of elementary abelian $p$-groups with $a \in A$ satisfying $C_B(a) = 1$, must $H = \langle a, B\rangle$ satisfy $\operatorname{rank}(Z(H) \cap H') \le \operatorname{rank}(B)$? An explicit extension violates the bound.
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