Geodesic
A question of Frohardt on $2$-groups, skew translation quadrangles of even order and cyclic STGQs
Forum of Mathematics, Sigma, Tome 11 (2023)

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We solve a fundamental question posed in Frohardt’s 1988 paper [6] on finite $2$-groups with Kantor familes, by showing that finite groups K with a Kantor family $(\mathcal {F},\mathcal {F}^*)$ having distinct members $A, B \in \mathcal {F}$ such that $A^* \cap B^*$ is a central subgroup of K and the quotient $K/(A^* \cap B^*)$ is abelian cannot exist if the center of K has exponent $4$ and the members of $\mathcal {F}$ are elementary abelian. Then we give a short geometrical proof of a recent result of Ott which says that finite skew translation quadrangles of even order $(t,t)$ (where t is not a square) are always translation generalized quadrangles. This is a consequence of a complete classification of finite cyclic skew translation quadrangles of order $(t,t)$ that we carry out in the present paper. 
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     author = {Koen Thas},
     title = {A question of {Frohardt} on $2$-groups, skew translation quadrangles of even order and cyclic {STGQs}},
     journal = {Forum of Mathematics, Sigma},
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     year = {2023},
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Koen Thas. A question of Frohardt on $2$-groups, skew translation quadrangles of even order and cyclic STGQs. Forum of Mathematics, Sigma, Tome 11 (2023). doi: 10.1017/fms.2023.105

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