The Maximum Distinguishing Number of a Group
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Let $G$ be a group acting faithfully on a set $X$. The distinguishing number of the action of $G$ on $X$, denoted $D_G(X)$, is the smallest number of colors such that there exists a coloring of $X$ where no nontrivial group element induces a color-preserving permutation of $X$. In this paper, we show that if $G$ is nilpotent of class $c$ or supersolvable of length $c$ then $G$ always acts with distinguishing number at most $c+1$. We obtain that all metacyclic groups act with distinguishing number at most 3; these include all groups of squarefree order. We also prove that the distinguishing number of the action of the general linear group $GL_n(K)$ over a field $K$ on the vector space $K^n$ is 2 if $K$ has at least $n+1$ elements.
1985 ◽
Vol 37
(2)
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pp. 238-259
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2005 ◽
Vol 15
(02)
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pp. 367-394
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1971 ◽
Vol 23
(4)
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pp. 679-685
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1988 ◽
Vol 43
(4)
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pp. 2533-2540
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2004 ◽
Vol 282
(1)
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pp. 368-385
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