Experimental Mathematics

Minimal Permutation Representations of Nilpotent Groups

Ben Elias, Lior Silberman, and Ramin Takloo-Bighash

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A minimal permutation representation of a finite group $G$ is a faithful $G$-set with the smallest possible size. We study the structure of such representations and show that for certain groups they may be obtained by a greedy construction. In these situations (except when central involutions intervene) all minimal permutation representations have the same set of orbit sizes. Using the same ideas, we also show that if the size $d(G)$ of a minimal faithful $G$-set is at least $c|G|$ for some $c>0$, then $d(G) = |G|/m + O(1)$ for an integer $m$, with the implied constant depending on $c$.

Article information

Experiment. Math. Volume 19, Issue 1 (2010), 121-128.

First available in Project Euclid: 12 March 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20B35: Subgroups of symmetric groups 20D15: Nilpotent groups, $p$-groups
Secondary: 20D30: Series and lattices of subgroups 20D60: Arithmetic and combinatorial problems

Permutation representations nilpotent groups lattices


Elias, Ben; Silberman, Lior; Takloo-Bighash, Ramin. Minimal Permutation Representations of Nilpotent Groups. Experiment. Math. 19 (2010), no. 1, 121--128. https://projecteuclid.org/euclid.em/1268404806.

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