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On a Galois
connection between the subfield lattice and the
multiplicative subgroup lattice
John K. McVey |
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Vol. 264 (2013), No. 1, 213–219
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Abstract |
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Given finite fields F < E, we present a collection of subgroups C ≤ E× and establish, to each C, a Galois connection between the intermediate field lattice ℰ = {L∣F ≤ L ≤ E} and C’s subgroup lattice. Our main result is that, in all but an extremely limited and completely determined family, the closed subset of ℰ is ℰ itself, establishing a natural bijection between ℰ and the lattice {L ∩ C∣L ∈ℰ}. As an application, we use this bijection to calculate the set of degrees for the complex-valued irreducible representations of the split extension C ⋊ Gal(E∕F). |
Keywords
Galois correspondence, lattice, character degree, finite
field
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Mathematical Subject Classification 2010
Primary: 06A15
Secondary: 20C15
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Milestones
Received: 2 June 2012
Revised: 10 December 2012
Accepted: 26 December 2012
Published: 5 July 2013
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Authors |
| John K. McVey | |
| Department of Mathematics Kent State University Mathematics and Computer Science Building 233 Summit Street Kent, OH 44242 United States |
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