Characterization of intersecting families of maximum size in \(\mathrm{PSL}(2,q)\). (English) Zbl 1385.05070
Summary: We consider the action of the 2-dimensional projective special linear group \(\mathrm{PSL}(2, q)\) on the projective line \(\mathrm{PG}(1, q)\) over the finite field \(\mathbb{F}_q\), where \(q\) is an odd prime power. A subset \(S\) of \(\mathrm{PSL}(2, q)\) is said to be an intersecting family if for any \(g_1, g_2 \in S\), there exists an element \(x \in \mathrm{PG}(1, q)\) such that \(x^{g_1} = x^{g_2}\). It is known that the maximum size of an intersecting family in \(\mathrm{PSL}(2, q)\) is \(q(q - 1) / 2\). We prove that all intersecting families of maximum size are cosets of point stabilizers for all odd prime powers \(q > 3\).
MSC:
| 05D05 | Extremal set theory |
| 20B30 | Symmetric groups |
| 20D06 | Simple groups: alternating groups and groups of Lie type |
Keywords:
character table; Erdős-Ko-Rado theorem; hypergeometric function over finite field; intersecting family; Legendre sum; Soto-Andrade sumReferences:
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